1
The function \(p(x)=-3x+4\) is defined for all real numbers. Which table shows values of \(x\) and \(p(x)\) correctly?
Linear functions: worksheet with answers
Work through the questions first, then open the answer key and explanations to check your work.
2
The function \(g(t)=7t-9\). What value of \(t\) satisfies \(g(t)=26\)?
3
If \(p(t)=8t-5\), what is the value of \(p(-4)\)?
4
A bookstore uses the function \(C(n)=9n+6\) to represent the total cost, in dollars, of buying \(n\) notebooks. What is the total cost when 8 notebooks are purchased?
5
A printing company charges a setup fee of $18 for each custom banner order and an additional $6 per banner printed. What is the total cost, in dollars, for an order of 14 banners?
6
A linear relationship between variables \(p\) and \(q\) has the following property: for every increase of 2 in \(p\), \(q\) decreases by 8. The graph of the relationship passes through the point \((3,11)\). Which equation represents this relationship?
7
The graph shows the total amount paid, in dollars, for using an equipment rental service for a certain number of hours. The company charges a fixed fee plus the same amount for each hour of use.
What does the slope of the line represent in this situation?
8
The equation \(4x-5y=15\) defines line \(m\). Line \(n\) is parallel to line \(m\). What is the slope of line \(n\)?
9
The graph of \(y=f(x)\) in the \(xy\)-plane represents the function \(f(x)=-3x+12\). Which choice best describes the meaning of the x-intercept of the graph?
10
A catering company tracks the number of plastic utensils it uses for events. The relationship between the total number of utensils used, \(u\), and the number of meal trays prepared, \(t\), is modeled by the equation \(u-4t=18\).
In this context, what does the number 18 represent?
11
A community kitchen prepares boxed lunches for events. The relationship \(p-8b=14\) models the situation, where \(p\) is the total number of sandwich rolls the kitchen has and \(b\) is the number of boxed lunches prepared. Each boxed lunch uses 8 sandwich rolls.
Which choice best describes the meaning of 14 in this context?
12
A kayak rental company charges $38 for the first hour of a rental and $12 for each additional hour. The function \(T(h)\) represents the total cost, in dollars, of renting a kayak for \(h\) hours, where \(h\) is a positive integer. Which function models this relationship?
Answer key and explanations
1. Answer
| x | p(x) |
|---|---|
| -2 | 10 |
| 0 | 4 |
| 3 | -5 |
Evaluate the function \(p(x)=-3x+4\) at each given value of \(x\).
For \(x=-2\): \(p(-2)=-3(-2)+4=6+4=10\).
For \(x=0\): \(p(0)=-3(0)+4=4\).
For \(x=3\): \(p(3)=-3(3)+4=-9+4=-5\).
The table with outputs 10, 4, and -5 is choice C.
2. Answer
5
Substitute 26 for \(g(t)\) to form the equation \(26=7t-9\). Add 9 to both sides to get \(35=7t\). Then divide both sides by 7:
\(t=5\)
3. Answer
-37
Substitute \(-4\) for \(t\) in the function.
\(p(-4)=8(-4)-5\)
Multiply first: \(8(-4)=-32\).
Then subtract 5: \(-32-5=-37\).
So the correct answer is \(-37\).
4. Answer
78
Substitute 8 for \(n\) in the function: \(C(8)=9(8)+6\). Then calculate \(9\cdot 8=72\), and add 6: \(72+6=78\). The total cost is 78 dollars.
5. Answer
102
The total cost can be modeled with a linear expression: \(18+6b\), where \(b\) is the number of banners. Substitute \(14\) for \(b\):
\(18+6(14)=18+84=102\)
So the total cost is $102.
6. Answer
\(q=-4p+23\)
If \(p\) increases by 2 while \(q\) decreases by 8, then the slope is \(\frac{-8}{2}=-4\). So the equation has the form \(q=-4p+b\).
Substitute the point \((3,11)\) into the equation:
\(11=-4(3)+b\)
\(11=-12+b\)
\(b=23\)
The equation is \(q=-4p+23\).
7. Answer
The amount charged for each additional hour of equipment use
In a linear graph, the slope represents how much the dependent variable changes for each 1-unit increase in the independent variable. Here, the independent variable is the number of hours, and the dependent variable is the total amount paid. Therefore, the slope represents the amount charged for each additional hour of equipment use.
Choice B is correct.
8. Answer
4/5
Rewrite the equation of line \(m\) in slope-intercept form:
\(4x-5y=15\)
\(-5y=-4x+15\)
\(y=\frac{4}{5}x-3\)
The slope of line \(m\) is \(\frac{4}{5}\). Parallel lines have the same slope, so the slope of line \(n\) is \(\frac{4}{5}\).
9. Answer
The output is 0 when the input is 4.
The x-intercept of a graph occurs where the output value is 0, so set \(f(x)=0\).
\(-3x+12=0\)
\(-3x=-12\)
\(x=4\)
This means the graph crosses the x-axis when the input is 4 and the output is 0. Therefore, the correct answer is choice C.
10. Answer
The company uses 18 extra utensils in addition to the utensils packed with the meal trays.
In the equation \(u-4t=18\), the term \(4t\) represents the number of utensils used for the meal trays because each tray requires 4 utensils. The equation says that after subtracting those utensils from the total number used, 18 utensils remain. Therefore, 18 represents the additional utensils used beyond the ones packed with the meal trays.
11. Answer
The kitchen has 14 extra sandwich rolls after making the boxed lunches.
In the equation \(p-8b=14\), the term \(8b\) represents the number of sandwich rolls used for the boxed lunches because each lunch uses 8 rolls. The value 14 is what remains after those rolls are used. Therefore, 14 represents the extra sandwich rolls the kitchen has beyond the number needed for the boxed lunches.
12. Answer
\(T(h)=12h+26\)
The first hour costs $38. For a rental of \(h\) hours, there are \(h-1\) additional hours that each cost $12. This gives the expression \(T(h)=38+12(h-1)\). Simplifying:
\(T(h)=38+12h-12=12h+26\)
So the correct model is \(T(h)=12h+26\).