# Graphing quadratic functions on the SAT

## SAT Subscore: Passport to Advanced Mathematics

### Graphing quadratic functions topic includes two parts

**Part 1- The features of the graphs of the quadratic functions**– x intercepts, y intercept, vertex points, vertical symmetry and width.

**Part 2- Three types of transformations of the parabola in the xy plane:**

Parabola translation– Shifting the graph up, down, to the left or to the right without changing its width, so that the distance that each point moves is the same.

Parabola translation is done by adding or subtracting a constant c to the function or from the x variable

Parabola stretching– Changing the graph’s width leaving the x intercepts coordinates and the axis of symmetry of the parabolas the same (the distance that each point moves is not the same).

Parabola stretching is done by multiplying the function by a constant:

Parabola reflecting– Reflecting the graph across the x axis or across the y axis.

Parabola reflecting is done by multiplying the function or the x variable by -1.

### Graphing quadratic functions versus linear functions

Graphing a quadratic function is different from graphing a linear function.

As we learned the graph of a linear function y=mx+b is graphed as a straight line. The x^{2 }term that is added in the quadratic function changes the graph to a form of a U shaped symmetrical curve. This graph of the quadratic function is called a parabola and has a vertex (a minimum or a maximum) and a y intercept.

The graph below shows the graph of a quadratic function y=x²+2x+10 and a linear function y=2x+10.

The difference between the functions is only the x² term. The table near the graphs shows different (x,y) values that were taken to plot the graphs. The only common point between the two graphs is when x=0, since then the term x² is equal to zero and does not affect the position of the parabola. This point (0,10) is also the y axis intercept.

### The steps for graphing quadratic functions:

When we want to graph a linear function, we need to draw a straight line between only 2 points. When we graph a quadratic function, we need more points since we need to cover the vertex, the decreasing part and the increasing part.

**The steps for graphing a quadratic function are:**

Step 1: Plug some x values into the function to calculate their corresponding y values.

Step 2: Plot the (x,y) coordinates in the xy plane.

Step 3: Sketch a parabola through all the points.

### The features of the graphs of quadratic functions:

#### Y intercept of a parabola

- Y intercept this is the point where the parabola crosses the y axis.
- Each parabola has a y intercept.
- There is only one y intercept for each parabola.
- In order to find the y intercept given the standard form of a quadratic function, we set x=0 and solve for y.

Consider the following example:

Graph the y intercept of a function y=x²+2x+10.

In order to find the y intercept, we set x=0 and solve for y:

y= x²+2x+10

y= 0²+2*0+10=10

The graph below shows the function y=x²+2x+10.

#### X intercept of a parabola

- X intercept this is the point where the parabola crosses the x axis.
- A parabola can have 0-2 x intercepts.
- X intercept is also called zero and root.

**We can see the x intercepts from the factored form of a quadratic function**. Given the factored form of y=a(x-b)(x-c), the intercepts are (b,0) and (c,0). For more details learn about finding x intercepts from a factored form of a quadratic equation topic.

**In order to find the x intercepts given the standard form of a quadratic equation** y= ax

^{2}+bx+c, we set y=0 and solve the quadratic equation ax

^{2}+bx+c=0 for x:

- If the equation has no real solution, then the parabola has no x intercepts.
- If the equation has 1 real solution, then the parabola has 1 x intercepts.
- If the equation has 2 real solutions, then the parabola has 2 x intercepts.

For more details about solving a quadratic equation learn the finding x intercepts from a factored form of a quadratic equation topic.

The graphs below show 3 functions __with a positive x__^{2}__ term (a>0)__:

y=x²+2x+1, y=2x²+x+1 and y=x²-2x-8.

The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The x intercept points are marked in red inside the table and on the graphs.

__The equation y=2x²+x+1=0__ has no solution, you can see that its graph y=2x²+x+1 is located above the x axis, therefore it has no x intercepts.

__The equation y=x²+2x+1=0__ has 1 solution, you can see that its graph y=x²+2x+1 intersects the x axis at 1 point (-1,0).

__The equation y=x²-2x-8=0__ has 2 solutions, you can see that its graph y= x²-2x-8 intersects the x axis at 2 points (-4,0) and (2,0).

See solving of these equations above in solving a quadratic equation topic.

The graphs below show 3 functions __with a negative x ^{2} term (a<0)__:

y=-x²+2x-5, y=-x²+2x-1 and y=-x²+2x+8.

The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The x intercept points are marked in red inside the table and on the graphs.

The equation y=-x²+2x-5 has no x intercepts.

The equation y=-x²+2x-1 intersects the x axis at 1 point (1,0).

The equation y=-x²+2x+8 intersects the x axis at 2 points (-2,0) and (4,0).

#### A vertex point of a parabola

- The vertex is the lowest point (
**minimum point**) or the highest point (**maximum point**) of the parabola. - A vertex is also called an extreme point.
- A parabola with a positive x
^{2}coefficient (a>0) has**1 minimum point.**The minimum is the point where the graph changes from decreasing to increasing. The shape of the parabola is convex, and it opens upward. - A parabola with a negative x
^{2}coefficient (a<0) has**1 maximum point.**The maximum is the point where the graph changes from increasing to decreasing. The shape of the parabola is concave, and it opens downward. - We can find the vertex of the parabola from its vertex function y=a(x-h)²+k, where the vertex coordinates are (h,k). Click here for more details about finding the vertex form of a quadratic function.

The graphs below show 2 functions: y=x²+2x+10 and y=-x²+2x+20.

The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The vertex points are marked in red and green inside the table and on the graphs.

The function y=x²+2x+10 has a vertex form y=(x+1)²+9. It has a positive x^{2} coefficient (a=1), a convex shape and a minimum point at (-1,9).

The function y=-x²+2x+20 has a vertex form y=-(x-1)²+21. It has a negative x^{2} coefficient (a=-1), a concave shape and a maximum point at (1,21).

#### A vertical symmetry of a parabola

**All parabolas are symmetric:** if we pass a vertical line through the vertex of the parabola, we will see that the right half is symmetric to the left half.

Note that since the parabola is symmetric, **when we move the same distance from the x coordinate of the vertex, we get the same y values in both points.**

For example: In the function y=x²+2x+10 the vertex is (-1,9). If we move to the right on the x axis to the point x=-1+1=0 we get y= 0²+2*0+10=10. If we move to the left on the x axis to the point x=-1-1=-2 we get y=-2²+2*-2+10=4-4+10=10. We got the same y value y=10.

The graphs below show 2 functions: y=x²+2x+10 and y=-x²+2x+20. One half of each parabola is colored in gray so you can see that the right half is symmetric to the left half.

The tables near the graphs show different (x,y) values that were taken to plot the graphs.

Some examples of vertex points are marked in colors inside the tables and on the graphs.

#### The width of a parabola- wide parabolas

The x^{2} coefficient (a) determines the width of the parabola: **The closer the value of x ^{2} coefficient (a) to 0, the wider the parabola. **

If x^{2} coefficient (a) is very close to 0, the function of the parabola is almost like a straight line (x^{2}*0=0), therefore the parabola is very wide.

The graphs below show 3 functions: y=2x+10, y=0.1x²+2x+10 and y=0.5x²+2x+10:

- The function y=2x+10 is a linear function and its graph is a straight line.
- The function y=0.1x²+2x+10 is a quadratic function in which we added to the linear function y=2x+10 a term of 0.1x². Since the coefficient of x² a=0.1 that is very close to 0, the parabola is very close to the graph of the straight line.
- The function y=0.5x²+2x+10 is a quadratic function in which we added to the linear function y=2x+10 x² a term of 0.5x².

Since 0.5x²>0.1x² the distance from the straight line of the function y=0.5x²+2x+10 is bigger than distance of the function y=0.1x²+2x+10.

The table near the graphs shows different (x,y) values that were taken to plot the graphs.

The point (0,10) marked in green is the point of the intersection between the graphs, since in this point a=0 and the parabolas become straight lines.

__A bigger x² coefficient (a) resulted in a bidder y value and a steeper parabola:__

You can see from the table and the graph that for every point from x=0 the y values of the blue function with a=0.1 are smaller than the y values of the orange function with a=0.5.

The example is marked in red on the graphs and in the table. At the point x=-5 the blue function with a=0.1 equals to y=0.1*-5²+2*-5+10=0.1*25-10+10=2.5.

The orange function with a=0.5 equals to y=0.5*-5²+2*-5+10=0.5*25-10+10=12.5.

What made the difference between the y values is 0.5*25=12.5 compared to 0.1*25=2.5.

The black arrow on the graph shows the distance between 12.5-2.5.

__The closer the value of x__^{2}__ coefficient (a) to 0, the closer the parabola to the straight line:__

You can see from the table and the graph that for every point from x=0 the y values of the blue function are closer to the y values of the grey straight line than the y values of the orange function with a=0.5. Therefore, the blue parabola with a=0.1 is closer to the grey straight line than the orange parabola with a=0.5.

#### The width of a parabola- narrow parabolas

**A large absolute value of x**Parabolas with large negative or positive coefficients (absolute values) are far from 0 and therefore narrow. In other words: parabolas with large magnitudes (absolute values of the coefficients) of x

^{2}coefficient (a) makes a steep and a narrow parabola.^{2}(a) are steeper and narrower than parabolas with small magnitudes that are wide.

The graphs below show 4 functions with ** positive x^{2} coefficients**: f(x)=x²+3x-10, f(x)=2x²+3x-10, y=3x²+3x-10 and y=4x²+3x-10. The only difference between the functions is the value of the x

^{2}coefficient (a) that becomes bigger.

The table near the graph shows different (x,y) values that were taken to plot the graphs.

__The bigger the value of the x ^{2} coefficient (a), the steeper is the parabola:__

You can see from the table and the graph that for every point from x=0 the y values of the blue function with a=1 are smaller than the y values of the orange function with a=2 and so on for the functions with a=3 and a=4. Bigger y values result in bigger distances from a straight line (a=0) and a steeper parabola.

The example is marked in red on the graphs and in the table. At the point x=2 the blue function with a=1 equals to f(x)=x²+3x-10=2²+3*2-10=4+6-10=0.

The orange function with a=2 equals to f(x)=2x²+3x-10=2*2²+3*2-10=8+6-10=4.

The grey function with a=3 equals to f(x)=3x²+3x-10=3*2²+3*2-10=12+6-10=8.

The yellow function with a=4 equals to f(x)=4x²+3x-10=4*2²+3*2-10=16+6-10=12.

What made the difference between the y values is 1*2² compared to 2*2² and so on.

The black arrow on the graph shows the distance between the y values of 0, 4, 8 and 12.

The graphs below show 4 functions with ** negative x^{2} coefficients**: f(x)=-x²+3x+10, f(x)=-2x²+3x+10, y=-3x²+3x+10 and y=-4x²+3x+10. The only difference between the functions is the value of the x

^{2}coefficient (a) that becomes bigger in absolute value (magnitude): -1, -2, -3 and -4.

The table near the graph shows different (x,y) values that were taken to plot the graphs.

__The bigger the absolute value of the x__^{2} (a) negati__ve coefficient, the steeper the parabola:__

You can see from the table and the graph that for every point from x=0 the y values of the orange function with a=1 are smaller (bigger in absolute value) than the y values of the blue function with a=2 and so on for the functions with a=3 and a=4. Smaller (bigger in absolute value) y values result in bigger distances from a straight line (a=0) and a steeper parabola.

### Transforming graphs of quadratic functions:

Transforming parabolas can be done by changing their functions so that different types of changes cause different types of transformations. There are 3 types of parabola transformations:

__Parabola translation__– moving the graph up, down, to the left or to the right without changing its width, so that the distance that each point moves is the same.__Parabola stretching__– changing the graph’s width, so that the distance that each point moves is not the same (leaving only the x intercepts coordinates the same).__Parabola reflecting__– reflecting the graph across the x axis or across the y axis.

### Translating the parabola in the xy plane:

We can make 4 translations to the graph y=f(x).

Note that the distance that each point moves must be the same.

**Shifting the parabola up or down:**

In this case x values don’t change and y values change, therefore we need to change the value of y.

__Shifting the parabola up__– the graph will move up so that the y value for every x will be bigger. This is possible by adding a constant c to the function, so that**y=f(x)+c.**__Shifting the parabola down__– the graph will move down so that the y value for every x will be smaller. This is possible by subtracting a constant c from the function, so that**y=f(x)-c.**

** ****Shifting the parabola to the left or to the right:**

In this case y values don’t change and x values change, therefore we need to change the value of x.

__Shifting the parabola to the left__– the graph will move to he left so that the x value for every y will be smaller. This is possible by adding a constant c to x, so that**y=f(x+c).**__Shifting the parabola to the right__– the graph will move to he right so that the x value for every y will be bigger. This is possible by subtracting a constant c from x, so that**y=f(x-c).**

#### Shifting the parabola up- adding a constant c to the function, so that y=f(x)+c:

We want the graph to move up, so that the y value for every x will be bigger (the distance that each point moves must be the same).

This is possible by adding a constant c to the function, so that y=f(x)+c. For example: shifting 2 points up will make the point (1,2) to be (1,4) and the point (3,4) to be (3,6).

Consider the following example:

Transform the graphs y=-x^{2}+2x-5 and y=x²-4x+8 four points up.

The transformed graph of y=f(x) is y=f(x)+c therefore:

The graph y=-x^{2}+2x-5 will become y=-x^{2}+2x-5+4=-x^{2}+2x-1

The graph y=x²-4x+8 will become y=x²-4x+8+4=x²-4x+12

The graphs below show the graphs of the given functions and their shifted graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The y values of each point in the shifted graphs are equal to the y values of the given graphs plus 4.

The red arrows near the graphs show the direction and the size of the movement- 4 points up.

As example, the points x=1 and x=3 are marked in brown and in purple in the tables and the graphs.

#### Shifting the parabola down- subtracting a constant c from the function, so that y=f(x)-c:

We want the graph to move down, so that the y value for every x will be smaller (the distance that each point moves must be the same).

This is possible by subtracting a constant c from the function, so that y=f(x)-c. For example: shifting 2 points down will make the point (1,2) to be (1,0) and the point (3,4) to be (3,2).

Consider the following example:

Transform the graphs y=-x^{2}+2x-1 and y=x²-4x+12 four points down.

The transformed graph of y=f(x) is y=f(x)+c therefore:

The graph y=-x^{2}+2x-1 will become y=-x^{2}+2x-1-4=-x^{2}+2x-5

The graph y=x²-4x+12 will become y=x²-4x+12-4=x²-4x+8

The graphs below show the graphs of the given functions and their shifted graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The y values of each point in the shifted graphs are equal to the y values of the given graphs minus 4.

The red arrows near the graphs show the direction and the size of the movement- 4 points down.

As example, the points x=2 and x=4 are marked in green and in purple in the tables and the graphs.

#### Shifting the parabola to the left - adding a constant c to the function, so that y=f(x+c):

The graph will move to he left so that the x value for every y will be smaller. Since y values do not change and x values change, this is possible by adding a constant c to x, so that y=f(x+c). For example: shifting 3 points to the left will change the point (1,2) to (-2,2) and the point (3,4) will change to (0,4).

__Why do we add a value to x?__

As we see in the example, for the function to move 3 units to the left each x value should become 3 units smaller. We want to plug x that is 3 units smaller into the transformed function and get the same y value as in the given function. In order to do that we need to add 3 units to x so it will become identical to the given x otherwise we cannot get the same y value (x-3+3=x). Therefore, if we take the transformed function y=f(x+3) and plug x values that are 3 units smaller we will get the original y values for every shifted x so that y=f(x+3-3)=f(x).

Consider the following example:

Transform the graphs y=-x^{2}+2x-5 and y=x²-4x+8 three points to the left.

The transformed graph of y=f(x) is y=f(x)+c, therefore the graph y=-x^{2}+2x-5 will become y=-(x**+3**)^{2}+2(x**+3**)-5

Checking the functions:

Plugging x=1 in the original function: y=-x^{2}+2x-5=-1+2-5=-4

Plugging x=1-3=-2 in the transformed function: y=-(x**+3**)^{2}+2(x**+3**)-5=-(-2**+3**)^{2}+2(-2**+3**)-5=-1+2-5=-4.

The y is the same (y=-4) in both functions and x moved 3 points to the left.

The transformed graph of y=f(x) is y=f(x)+c, therefore the graph y=x²-4x+8 will become y=(x**+3**)²-4(x**+3**)+8

Checking the functions:

Plugging x=1 in the original function: y=x²-4x+8=1-4+8=5

Plugging x=1-3=-2 in the transformed function: y=(x**+3**)²-4(x**+3**)+8=1-4+8=5

The y is the same (y=5) in both functions and x moved 3 points to the left.

The graphs below show the graphs of the given functions and their shifted graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The x values of each point in the shifted graphs are equal to the x values of the given graphs minus 3.

The red arrows near the graphs show the direction and the size of the movement- 3 points to the left.

As example, the points x=2 and x=-1 are marked in green and the point x=1 and x=4 are marked in purple in the tables and the graphs.

#### Shifting the parabola to the right - subtracting a constant c from the function, so that y=f(x-c):

The graph will move to he right so that the x value for every y will be bigger. Since y values don’t change and x values change, this is possible by subtracting a constant c from x, so that y=f(x-c). For example: shifting 3 points to the right will change the point (1,2) to (4,2) and the point (3,4) will change to (6,4).

__Why do we subtract a value from x?__

As we see in the example, for the function to move 3 units to the right each x value should become 3 units bigger. We want to plug x that is 3 units bigger into the transformed function and get the same y value as in the given function. In order to do that we need to subtract 3 units from x so it will become identical to the given x otherwise we can’t get the same y value (x+3-3=x). Therefore, if we take the transformed function y=f(x-3) and plug x values that are 3 units bigger we will get the original y values for every shifted x so that y=f(x-3+3)=f(x).

Consider the following example:

Transform the graphs y=-x^{2}+2x-5 and y=x²-4x+8 three points to the right.

The transformed graph of y=f(x) is y=f(x)-c, therefore the graph y=-x^{2}+2x-5 will become y=-(x**-3**)^{2}+2(x**-3**)-5.

Checking the functions:

Plugging x=1 in the original function: y=-x^{2}+2x-5=-1+2-5=-4

Plugging x=1+3=4 in the transformed function: y=-(x**-3**)^{2}+2(x**-3**)-5=-(4**-3**)^{2}+2(4**-3**)-5=-1+2-5=-4.

The y is the same (y=-4) in both functions and x moved 3 points to the right.

The transformed graph of y=f(x) is y=f(x)-c, therefore the graph y=x²-4x+8 will become y=(x**-3**)²-4(x**-3**)+8.

Checking the functions:

Plugging x=1 in the original function: y=x²-4x+8=1-4+8=5

Plugging x=1+3=4 in the transformed function: y=(x**-3**)²-4(x**-3**)+8=1-4+8=5

The y is the same (y=5) in both functions and x moved 3 points to the right.

The graphs below show the graphs of the given functions and their shifted graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The x values of each point in the shifted graphs are equal to the x values of the given graphs plus 3.

The red arrows near the graphs show the direction and the size of the movement- 3 points to the right.

As example, the points x=1 and x=4 are marked in green and the point x=2 and x=5 are marked in purple in the tables and the graphs.

### Stretching the parabola vertically in the xy plane:

#### Stretching by multiplying the function by a constant c>1

To stretch the graph vertically we need **to multiply the function by a constant c>1.** Stretching the graph y=f(x) vertically by a factor of c will give us a new graph y=f(x)*c.

**If the graph is located above the x axis** (all y values are positive), y values in the stretched graph will be bigger than in the original graph, therefore the stretched graph will move up above the original graph. Y values are bigger because a positive y multiplied by a number bigger than 1 will get us a bigger y in the stretched graph.

**If the graph is located below the x axis** (all y values are positive), y values in the stretched graph will be smaller than in the original graph, therefore the stretched graph will move down below the original graph. Y values are smaller because a negative y multiplied by a number bigger than 1 will get us a smaller y in the stretched graph.

Consider the following example:

Sretch the graphs y=-x^{2}+2x-5 and y=x²-4x+8 vertically by a factor of 2.

The graphs below show the graphs of the given functions and their stretched graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The y values of each point in the stretched graphs are equal to the y values of the given graphs multiplied by 2.

The red arrows near the graphs show the direction of the movement. Note that the size of the movement changes between the points- bigger (in absolute values) y values result in bigger movements.

For example: the points x=0 and x=-2 are marked in green and purple in the tables and the graphs. The green point (0,-5) in the original graph stretched by 2 to the point (0,-10) in the stretched graph. The movement of the graph was -10 – -5= -10+5=-5 points down. The purple point (-2,-13) in the original graph stretched by 2 to the point (-2,-26) in the stretched graph. The movement of the graph was -26 – -13= -26+13=-13 points down.

**If the graph is located below and above the x axis **(the graph has 2 x intercepts), the movement of the stretched graph has 3 forms:

__The intersection with the x axis__ where y=0 are the same points in the both graphs. This is because if we multiply y=0 in the original graph we get y=0 in the stretched graph.

__The points above the x axis__ (where y is positive) will move up in the stretched graph so that the stretched graph will be above the original graph.

__The points below the x axis__ (where y is negative) will move down in the stretched graph so that the stretched graph will be below the original graph.

Consider the following example:

Stretch the graphs y=-x^{2}+2x-5 and y=x²-4x+8 by a factor of 2.

The graphs below show the graphs of the given functions and their stretched graphs.

The tables near the graphs show different (x,y) values that were taken to plot the graphs. The y values of each point in the stretched graphs are equal to the y values of the given graphs multiplied by 2.

The red arrows near the graphs show the direction of the movement. Note that the size of the movement changes between the points- bigger (in absolute values) y values result in bigger movements.

For example: the points x=-4 and x=0 are marked in green and purple in the tables and the graphs. The green point (0,14) in the original graph stretched by 2 to the point (0,28) in the stretched graph. The movement of the graph was 28-14=14 points up. The purple point (-4,-6) in the original graph stretched by 2 to the point (-4,-12) in the stretched graph. The movement of the graph was -12 – -6= -12+6=-6 points down.

#### Shrinking by multiplying the function by a constant 1>c>0

To shrink the graph vertically we need **to multiply the function by a constant 1>c>0 **Shrinking the graph y=f(x) vertically by a factor of c will give us a new graph y=f(x)*c. For example: the graph 0.5*f(x) is the graph f(x) shrunken by 2 (or stretched by 0.5).

Shrinking is the opposite to stretching:

**For graphs above the x axis** that have positive y values: if we multiply a positive y value by a number 1>c>0 we get a smaller y value, therefore the graph will move down.

**For graphs below the x axis** that have negative y values: if we multiply a negative y value by a number 1>c>0 we get a smaller negative y value, therefore the graph will move up.

**If the graph is located below and above the x axis **(the graph has 2 x intercepts), the movement of the stretched graph has 3 forms:__The intersection with the x axis__ where y=0 are the same points in the both graphs. This is because if we multiply y=0 in the original graph we get y=0 in the shrunken graph.__The points above the x axis__ (where y is positive) will move down in the shrunken graph so that the shrunken graph will be below the original graph.__The points below the x axis__ (where y is negative) will move up in the shrunken graph so that the shrunken graph will be above the original graph.

Consider the following example:

Shrink and stretch the graphs y=-x^{2}+2x-5 and y=x²-4x+8 by a factor of 2.

The graphs below show the graphs of the given functions and their stretched and shrunken graphs.

### Reflecting the parabola across the x axis and the y axis

#### A vertical reflection across the x axis

Reflecting the graph y=f(x) vertically will give us a new graph y=f(x)*-1.

Note that if we reflect the reflected graph again, we will get the original graph back, since y=f(x)*-1*-1=f(x). Meaning that the 2 graphs y=f(x) and y=-f(x) are reflection of each other.

Consider the following example:

Reflect across the x axis the parabola y=-x²+10x-21.

If the original graph is y=-x²+10x-21 then the reflected across the x axis graph will be

y=-1*(-x²+10x-21)

y=x²-10x+21

The graphs below show the graph of the given function and its reflected graph.

The black graph represents the original parabola and the orange graph represents the reflected parabola. The green arrows near the graphs show the direction of the reflection.

The table near the graphs shows different (x,y) values that were taken to plot the graphs. The y values of each point in the reflected graph are equal to the y values of the given graph multiplied by -1. As example, the point x=5 is marked in red in the tables and the graphs.

Note that the black graph is also a vertical reflection of the orange graph:

The orange graph: y=x²-10x+21

The reflection of the orange graph is y=-1(x²-10x+21)=-x²+10x-21, this is the black graph.

#### A horizontal reflection across the y axis

**A horizontal reflection** reflects a graph horizontally across the y axis, so that the reflected graph is a mirror image of the original graph about the y axis (the mirroring is on the left and on the right of the y axis).

In order to move the graph across the y axis we need to change the sign of x coordinate for every y coordinate. Meaning that in order to make a reflection across the y axis the y coordinate does not change, and the x coordinate changes its sign. For example: If the original point is (1,2) the reflected across the x axis point will be (-1, 2).

Reflecting the graph y=f(x) horizontally will give us a new graph y=f(x*-1).

Note that if we reflect the reflected graph again, we will get the original graph back, since y=f(x*-1*-1)=f(x). Meaning that the 2 graphs y=f(x) and y=f(-x) are reflection of each other.

Consider the following example:

Reflect across the x axis the parabola y=-x²+10x-21.

If the original graph is y=-x²+10x-21 then the reflected across the y axis graph will be

y=-(x*-1)²+10x*-1-21

y=-x²-10x-21

The graphs below show the graph of the given function and its reflected graph.

The black graph represents the original parabola and the green graph represents the reflected parabola. The blue arrows near the graphs show the direction of the reflection.

The table near the graphs shows different (x,y) values that were taken to plot the graphs. The x values of each point in the reflected graph are equal to the y values of the given graph multiplied by -1. As example, the point y=4 is marked in red in the tables and the graphs.

Note that the black graph is also a horizontal reflection of the green graph:

The green graph is y=-x²-10x-21.

The reflection of the green graph: y=-(x*-1)²-10x*-1-21=-x²+10x-21 this is the black graph.

#### A horizontal reflection across the y axis versus a vertical reflection across the x axis

The graphs below combine horizontal and vertical reflections of the graph of y=-x²+10x-21 explained earlier.

The black graph represents the original parabola.

The green graph represents the horizontal reflection of the parabola across the y axis.

The orange graph represents the vertical reflection of the parabola across the x axis.

The table near the graphs shows different (x,y) values that were taken to plot the graphs. As example, the point (5,4) and its reflected points are marked in red in the tables and the graphs. The point (5,4) was first reflected across the x axis to the point (5,-4) in the orange graph and then it was reflected across the y axis to the point (-5,4) in the green graph.