Linear Functions on the SAT Test
SAT Subscore: Heart of Algebra
Studying linear functions
On the SAT test linear functions are part of heart of algebra subscore that includes 4 fundamental topics that appear in many SAT questions. Start studying heart of algebra subscore with this topic and continue to the next 3 topics that include linear equations, systems of linear equations and linear inequalities.
Linear functions topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.
Linear functions- summary
A linear function is an equation that represents a relationship between two variables, most commonly called x and y. It is called linear because it can be graphed as a straight line in the xy-plane.
Creating a linear function requires writing a function that describes a given word problem without solving it.
Finding the value of a function requires solving a function given the values of its variables.
Finding the input that corresponds to a given output requires to calculate the input of a function given the value of the output.
Interpretation of a linear function includes analyzing a linear function in its slope-intercept form: y=mx+b (rewriting the function from its standard form ax+ty=c, calculating the slope and the intercept of the function and analyzing of the graph of the function).
Finding the equation of a linear function requires to write the equation of a linear function y=mx+b given the values of its points or its slope b.
Graphic presentation of a linear function requires identifying a graph of a given linear function or finding the equation of a linear function given its graph.
Analyzing relationships between two linear functions includes finding the slope of a linear function given the slope of another function or finding the equation of a linear function given the equation of another function.
The formula sheet for linear functions is listed below.
Linear functions formula sheet
The linear functions formula sheet is given below, it includes the following formulas:
Two forms in which we can write a linear function: the standard form ax+ty=c and the slope intercept form y=mx+b.
The formula for calculating the value of the slope of a linear function given 2 points a and b: m=(ya-yb)/(xa-xb).
Two formulas the represent the relationships between two linear functions: parallel lines slopes formula m1=m2 and perpendicular lines slopes formula m1*m2=-1.
Note that all the formulas are explained in detail on this page.
Creating a linear function
In these questions you are given a word problem and you are asked to write the function that describes the problem without solving it.
The skill required is translating a word problem into a mathematical equation.
Consider the following example:
The full price of the item is X dollars. During the holiday sale all the products in the store are sold at a twenty percent discount. In addition, there is a second discount of 10 percent on the item.
What is the function P that represents the price after the discounts in terms of x?
Case 1: The additional discounts are calculated on the price after the holiday discount.
Case 2: The additional discounts are calculated on the original price.
The price of the item after the discount is:
Case 1: P=x*(1-0.2)*(1-0.1)=x*0.8*0.9=x*0.72=0.72x
Case 2: P=x*(1-0.2-0.1)=x*0.7=0.7x
Note that the difference between the two answers is 10 percent from 20 percent: 0.1 times 0.2=0.02.
Finding the value of an output of a given function
In these questions you are given the equation of the function and you are asked to solve this function given the values of the variables.
The skills required are the ability to plug data instead of variables and making a simple calculation.
Solution steps: Data identification, plugging the data into the function, solving the expression.
Consider the following example:
The plant uses function c defined by c(f,n,v)=f/n+v to calculate the cost c(f,n,v), in dollars, of producing a product. F is defined as a fixed cost of the plant; v is defined as a variable production cost per unit and n is defined as the quantity of the products.
If the fixed cost is $ 50,000 and the variable cost is $ 1, what is the manufacturing cost per product in the production of 100,000 products?
In order to answer this question, we must plug the data into the function and solve it.
Step 1: writing the data:
c(f,n,v)=f/n+v
f=50,000
v=1
n=100,000
Step 2: Plugging the data into the function:
c(f=50000, n=100000, v=1)= 50,000/100,000+1
Step 3: Solving the expression:
50,000/100,000+1=0.5+1=1.5
The answer is that the manufacturing cost of 1 unit is 1.5 dollars.
Finding the input that corresponds to a given output of a function
In these questions you know the equation of the function and you are asked to calculate the input of this function given the value of the output.
The skills required are the ability to plug data instead of variables and solving the equation by isolating its input variable.
Solution steps: Plugging the data into the function, solving the equation.
Consider the following example:
The camp manager uses the function f(x) defined by f(x)=430+55*x to calculate the total number of the children f(x) in the summer camp. More children join the camp every day on x buses.
If there were 760 children in the camp today, how many buses arrived at the camp?
In this question we are given the output of the function which is 760 children. This allows us to write an equation from the function and find the variable x.
Step 1: Plugging the output into the function:
760=430+55x
Step 2: Solving the equation:
55x=330
x=6
The answer is that 6 buses arrived today.
Step 3: checking the answer:
In order to test the answer we will place it in the equation: 430+55*6=430+300+30=760.
Interpretation of a linear function
In these questions you need to analyze the linear function in its slope-intercept form: y=mx+b.
The skills required:
Understanding the structure of the function.
Rewriting the function in a slope-intercept form.
Performing calculations to find the slope m and the intercept b.
Finding the equation of the function.
Presentation and analysis of the graph of a linear function.
What is a slope-intercept form of a linear function?
Slope intercept form of a linear function:
y=mx+b
For example: y=4x-3
Standard form of a linear function:
ax+ty=c
For example: 3x+4y=20
A linear function is a linear equation with 2 variables. We often see it the standard form as ax+ty=c (where a ,t and c are constants). For example: The price of 3 pens and 4 pencils is 10 dollars 3x+4y=20.
It is important to know that a linear function can be written in a slope intercept form y=mx+b where x and y are variables and m and b are constants. This form is essential to understanding how variable x affects variable y and is used to draw the function on a xy-plane.
The lines in the plot below represent 3 linear functions with different slopes and intercepts.
The constant m represents the slope of the function:
If the slope m is positive then the line trends upward from left to right (see the blue line).
If the slope m is negative then the line trends downward from left to right (see the orange line).
If the slope m is equal to zero then the line is parallel to the x axis (see the red line).
The constant b represents the intercept of the function with y axis. This is the point where the line crosses the y axis. In order to find b, we can plug x=0 into the equation and find y.
We can also calculate the intercept of the line with the x axis by plugging y=0 into the equation and finding x.
How to identify the slope m and the intercept b in the function equation?
As said earlier we often see the standard form of a linear function as ax+ty=c (where a ,t and c are constants). Since the equation isn’t written as y=mx+b we need to rewrite it in slope-intercept form.
Consider this example:
What is the slope and the intercept of the function 6x-3y+12=0?
6x-3y+12=0
6x+12=3y
y=2x+4
m=2 is the slope and b=4 is the intercept.
Writing the equation of the linear function
In order to write an equation of a linear function we need to know its slope and intercept and plug them in the formula of the function: y=mx+b. Consider this example:
The slope of a linear function is 5, the intercept is -3. Write the equation of the linear function.
We are given that m=5 and b=-3 therefore the equation in a form of y=mx+b is y=5x-3.
How can we calculate the slope m?
Linear slope formula:
m=(ya-yb)/(xa-xb)
For example: For A(-2,-9) and B(0,-3) the slope is m=(-9–3)/(-2-0)=-6/-2=3
We can calculate the slope m by using the x and y values of any two points A and B that are on the graph of the function.
The slope m is the difference between the y values and divided by the difference in the x values of the 2 points. Since the function is a straight line the slope is fixed and does not change at any point selected in the function.
The formula for calculating a slope is: m=(ya-yb)/(xa-xb)
Note that if you write the y value of point A first, then you should also write the x value of point A first.
Consider this example:
There are 2 points on the graph of a linear function A(-2,-9), B(0,-3). What is the slope m of the function?
m=(ya-yb)/(xa-xb)=(-9–3)/(-2-0)=-6/-2=3. The answer is m=3.
Note that we can write first the values of point B and subtract from them the values of point A, the slope m will be the same: m=(-3–9)/(0–2)=6/2=3.
Finding the equation of a linear function
In these questions you need to write the equation of the linear function y=mx+b given the values of its points or its slope b.
The skills required:
Understanding the structure of the function.
Performing calculations to find the slope m and the intercept b.
Finding the equation of the function.
Solving an equation with one variable.
Solving an equation with two variables.
As we know the equation of a linear formula is y=mx+b. Without any other data we have 2 unknown variables (x and y) and 2 unknown constants (m and b). In order to write the equation, we need to find the value of m and b.
There are 2 ways we can find the equation:
1. If we are given x and y values of one point on the line and the value of the slope m. We can write an equation with only one variable which in b and solve it.
2. If we are given x and y values of two points on the line. We can write 2 equations with 2 variables m and b and solve them. Another option is to calculate the slope m and then write an equation with only one variable b and solve it.
Finding the equation of a linear function given the value of one point and the value of the slope m:
We know the slope so we only need to calculate the intercept b by plugging the m value and the point values into the function formula y=mx+b. Consider this example:
In a linear function: the slope m=3, there is a point on the function which values are A(-2,-9). What is the equation of the function?
The linear function formula is y=mx+b
-9=3*-2+b
-9=-6+b
b=-3
The function equation is therefore y=3x-3
Finding the equation of a linear function given x and y values of two points:
Option 1 for solution– Using the slope formula in the calculation:
Step 1: Calculating the slope m using the slope formula.
Step 2: Calculating the intercept b by placing the m value and the point values into the function formula y=mx+b
Option 2 for solution: Solving with a system of equations:
We can 2 write and solve 2 equations so that each equation represents a point and the variables are the slope m and the intercept b.
Consider this example:
There are 2 points on the graph of a linear function A(-2,-9), B(0,-3). What is the equation of the function?
Option 1 for solution:
Step 1: We had already calculated the slope m in the previous question as m=(ya-yb)/(xa-xb)=(-9–3)/(-2-0)=-6/-2=3.
Step 2: We had already calculated the intercept b in the previous question as
-9=3*-2+b
-9=-6+b
b=-3
Option 2 for solution:
Solving the equation system y=3x-3, y=-2x+5
3x-3=-2x+5
5x=8
x=8/5=1+3/5=1+0.6=1.6
y=3x-3=3*1.6-3=3+3*0.6-3=3*0.6=1.8
The solution is x=1.6 y=1.8 (1.6,1.8).
Graphic presentation of a linear function
In these questions you need to identify a graph of a given linear function or to find the equation of a linear function y=mx+b given the graph of the function.
The skills required:
Understanding the structure of a linear function.
Writing a linear function in a slope intercept form.
Graphing a linear function in the xy plane.
Finding the equation of the function:
Performing calculations to find the slope m and the intercept b.
Solving an equation with one variable.
Solving an equation with two variables.
The graph of a linear function is a line in the xy plane. We can graph linear equation by plugging different numbers into the equation. Note that you need to plug x=0 x in order to find the intercept with the y axis represented as m.
How can we draw a linear graph?
In order to draw the graph, we need to mark 2 points and draw a line between them. To make sure you have found the right points it is best to mark a third point. If there is no mistake all the points will be on the line.
Which points to choose? Mark the points of intersection with the x-axis (y=0) and intersection with the y-axis (x=0) and then select a third point that is located between these two points.
Finding the equation of a linear function given the graph of the function:
Sometimes no values are given and instead we see the graph of the function. How can we write the equation of the function using only it’s graph?
Step 1: Finding two points values:
For making an easy and accurate calculation we need to look for 2 points whose (x,y) values are not a fraction.
Step 2: Calculating the slope m value by plugging the points we found from the graph into the slope formula m=(ya-yb)/(xa-xb).
3. Step 3: Finding the function formula by plugging one of the points and the slope m we calculated into the function formula y=mx+b.
Consider this example:
What is the equation of the function in the graph?
Step 1: Finding two points values:
We can see that the y intercept has the value of 14 so we know one point (0,14). This point is also the intercept b from the line formula y=mx+b.
The x intercept is between 3 and 4 therefore we don’t know its exact value and we can’t take it as a point.
We can easily see the point (2,6) on the graph so we can use it.
Step 2: Calculating the slope m value by plugging the points (2,6) and (0,14) into the slope formula:
m=(ya-yb)/(xa-xb)=(6-14)/(2-0)=-8/2=-4
Step 3: Finding the function formula by plugging one of the points (2,6) or (0,14) and the slope m=-4 into the function formula y=mx+b:
6=-4*2+n
n=14
14=n+-4*0
n=14
Relationships between two linear functions
In these questions you need to find the slope of a linear function given the slope of another function or to find the equation of a linear function given an equation of another function.
The skills required:
Understanding the structure of a linear function.
Writing a linear function in a slope intercept form.
Finding the slope of one line by using the slope of a parallel line.
Finding the slope of one line by using the slope of a perpendicular line.
Finding the equation of the function:
Performing calculations to find the intercept b
Performing calculations to find the x value or y value of a point.
Solving an equation with one variable.
Finding the slope of a linear function given the slope of another function:
Parallel lines slope formula:
m1=m2
For example: if m1=-2 then m2=-2
Perpendicular lines slope formula:
m1*m2=-1
For example: if m1=-2 then m2=1/2
There are two cases when we can find the slope of one line by using the slope of another line: the lines are parallel or the lines are perpendicular.
Parallel lines have the same slope m, so if we know the slope of one line then the slope of the parallel line will be the same.
Perpendicular lines slopes form 90° angles at their cutting point.
The product of the slopes of vertical lines is equal to minus 1 meaning that
m1*m2=-1 (m1 is the slope of the first line and m2 is the slope of the second line).
Consider this example:
The equation of function a is 3y+9x-36=0. Function b is parallel to function a and function c is perpendicular to function a.
What are the slopes of the functions a ,b and c?
Step 1: We will write the function a in the form y=mx+b:
3y+9x-36=0
3y=-9x+36
y=-3x+12
Step 2: Finding the slope of the parallel function b:
The slope of function b equals to the slope of function a, therefore the slope of function b is -3.
Step 3: Finding the slope of the perpendicular function c:
We need to plug the slope of function a into the formula m1*m2=-1.
m1=-3
-3*m2=-1
m2=-1/-3
m2=1/3
The slope of function c is 1/3.
The graph shown below presents the functions.
Finding the equation of a linear function given an equation of another function:
The equation of function a is y=-3x+12. Function b is perpendicular to function a and intersects it in a point x=4 so that the angle between the functions is equal to 90 degrees.
Write the equation of function b.
Step 1: Finding the slope of function b:
From the equation of function a we know that its slope is m=-3.
We are given that function b is perpendicular to function a, therefore we need to plug the slope of function a into the formula m1*m2=-1.
-3*m2=-1
m2=-1/-3
m2=1/3
The slope of function b is 1/3.
Step 2: Finding the y value of the intersection point:
We know that the functions intersect at x=3. We can plug x=3 into function a and find the value of y.
y=-3x+12
y=-3*4+12
y=0
The intersection point is (-3,0).
Step 3: Finding the y-intercept b of function b:
We know the value of a point on function b and the value its slope, therefore we can find the value its intercept b.
y=mx+b
0=1⁄3*4+b
b=-4/3
The formula of function b is y=1⁄3x-4⁄3
Step 4: Checking the answer:
Multiply the slopes to see if the result is -1.
(1⁄3)*(-3)=(1*-3)/3=-3/3=-1
Plug the point (4,0) into each equation to see if you get y=0.
y=-3x+12=-3*4+12=0
y=1⁄3x+4⁄3=(1⁄3)*4-4⁄3=4⁄3–4⁄3=0
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