Functions
The relation {(−3, 7), (2, 5), (6, 9)} is a function. Give an ordered pair that would make it NOT a function if added.
Example: (2, 8), because the input 2 would have two different outputs.
Store A sells notebooks using the equation C = 2.50n. Store B sells 4 notebooks for $12. Which store is cheaper per notebook, and by how much?
Store A is cheaper by $0.50 per notebook. Store B charges $12 ÷ 4 = $3 per notebook.
A line passes through (1, 3) and (4, 15). Find its slope. If (c, 11) is also on the line, what is c?
The slope is 4, and c = 3. The line rises 4 units for every 1 unit to the right.
Classify each equation as linear or nonlinear:
y = x(2x + 5)
y = x ÷ 4 + 6
y = 9 − 2x.
y = x(2x + 5) is nonlinear because it simplifies to
y = 2x² + 5x.
The other two equations are linear.
Arcade A charges a $6 entry fee plus $2 per game. Arcade B charges a $12 entry fee plus $1 per game. After how many games will the total costs be equal, and what will that cost be?
6 games. Both arcades will cost $18.
What is a function?
A relationship in which each input has exactly one output.
What is the constant of proportionality?
How do you find it?
The constant ratio of y to x in a proportional relationship.
It is found using k = y ÷ x when x ≠ 0.
What is slope?
What are the three ways to find it .
A measure of a line’s steepness and direction. It is the change in y divided by the change in x.
Rise / Run
Change of y / Change in x
y2-y1 / x2-x1
What is a linear function?
A function with a constant rate of change whose graph is a straight line.
What is the y-intercept of a graph?
The point where the graph crosses the y-axis. At this point, x = 0.
Does {(1, 4), (2, 4), (3, 4), (4, 4)} represent a function? Explain.
Yes. Each input has exactly one output. Different inputs can have the same output.
A proportional relationship contains the points (2, 14), (4, 28), and (6, 42). Find the constant of proportionality and write the equation.
k = 7
y = 7x
Moving from left to right, a line goes down 6 units and right 3 units. What is its slope?
−6 ÷ 3 = −2
A table has the ordered pairs (0, 1), (1, 3), (2, 7), and (3, 13). Could these points represent a linear function? Explain.
No. As x increases by 1, y increases by 2, then 4, then 6. The rate of change is not constant.
A bike rental shop charges a $9 starting fee plus $4 per hour. Write an equation for the total cost, C, after h hours.
C = 4h + 9
What is the vertical line test?
A test used to determine whether a graph represents a function. If any vertical line crosses the graph more than once, it is not a function.
What is a proportional relationship?
A relationship in which one quantity is always a constant multiple of the other. It can be written as y = kx.
A line contains the points (0, 8), (2, 14), and (4, 20). What is its slope?
3, because the change in y is 6 and the change in x is 2. So, 6 ÷ 2 = 3.
What is a nonlinear function?
A function whose graph is not a straight line. Its rate of change is not constant.
DJ A charges according to C = 35h + 40. DJ B charges a starting fee and a constant hourly rate: $110 for 2 hours and $170 for 4 hours. Which DJ has the higher hourly rate, and by how much?
DJ A: $35 per hour (the coefficient of \(h\)).
DJ B: \(\frac{170-110}{4-2}=\frac{60}{2}=\$30\) per hour.
DJ A has the higher hourly rate by $5 per hour.
A vertical line crosses a graph at (3, 2) and (3, −4). Does the graph represent a function? Explain.
No. The input 3 has two different outputs, so the graph fails the vertical line test
A graph is a straight line that passes through (0, 5). Does it represent a proportional relationship? Explain.
No. A proportional relationship must have a straight-line graph that passes through the origin, (0, 0).
What type of slope does a horizontal line have? What type does a vertical line have?
A horizontal line has zero slope. A vertical line has undefined slope.
A student says, “y = 4x + 7 is nonlinear because its graph does not pass through the origin.” Is the student correct? Explain.
No. The equation is linear because its graph is a straight line with a constant slope of 4. A linear function does not have to pass through the origin.
A water tank starts with 18 gallons and fills at 5 gallons per minute. Write an equation for the amount of water, W, after t minutes.
W = 5t + 18