8.PAR.4.1
A proportional relationship contains the points (2, 16), (4, 32), and (6, 48). Write an equation that represents the relationship.
y = 8x
Determine whether (3, 7) is a solution to y = 2x + 1. Show how you know.
Yes. 2(3) + 1 = 7, so (3, 7) is a solution.
Does the relation {(2, 4), (3, 4), (4, 4)} represent a function? Explain why or why not.
Yes. Each input has exactly one output.
Is the equation y = 4x - 3 linear or nonlinear? Explain how you know.
Linear. It is in the form y = mx + b.
A movie theater charges $9 for each ticket plus a one-time $4 online fee. Write an expression for the total cost of x tickets.
9x + 4
A gym charges $40 after 1 month, $65 after 2 months, $90 after 3 months, and $115 after 4 months. Write an equation for the total cost y after x months. Then state whether the relationship is proportional or non-proportional.
y = 25x + 15; non-proportional
Find the y-coordinate of the solution to y = -3x + 7 when x = 4.
y = -5
The input 2 maps to 7, the input 3 maps to 7, and the input 4 maps to 7. Does this mapping represent a function? Explain.
Yes. Different inputs can have the same output. Each input still has exactly one output.
Is the equation y = x² + 2 linear or nonlinear? Explain how you know.
Nonlinear. The variable x has an exponent of 2, so the graph is not a straight line.
Solve:
3x + 7 = 22
x = 5
The equation y = 7x represents a relationship. Explain how you know the relationship is proportional.
It is proportional because it is in the form y = kx, has a y-intercept of 0, and passes through the origin.
Give three ordered pairs that are solutions to y = x - 2.
Answers vary.
(0, -2), (1, -1), (3, 1)
The relation {(1, 2), (1, 6), (3, 8)} does not represent a function. Explain exactly why.
The input 1 has two different outputs, 2 and 6.
A function contains the points (0, 2), (1, 5), (2, 8), and (3, 11). Is the function linear or nonlinear? Explain how you know.
Linear. The y-values increase by 3 each time x increases by 1, so the rate of change is constant.
Solve the inequality:
4 - 3x ≥ 19
x ≤ -5
A proportional graph contains the points (0, 0), (5, 35), (10, 70), and (15, 105). Find the constant of proportionality (Slope) and write the equation of the graph.
Constant of proportionality (slope) = 7; y = 7x
The point (5, y) is a solution to y = 4x - 3. Find y and explain how you found it.
y = 17. Substitute x = 5: y = 4(5) - 3 = 17.
A graph is a circle. Explain how the vertical line test shows that the graph is not a function.
Some vertical lines intersect the circle more than once, meaning one x-value has more than one y-value.
A function contains the points (0, 1), (1, 3), (2, 7), and (3, 13). Is the function linear or nonlinear? Explain how you know.
Nonlinear. The changes in the y-values are +2, +4, and +6, so the rate of change is not constant.
Determine whether the equation has one solution, no solution, or infinitely many solutions.
4(x + 2) = 4x + 8
Infinitely many solutions
A delivery service charges $18 for 2 miles, $30 for 4 miles, and $42 for 6 miles. Determine whether the relationship is proportional. If it is not, write a linear equation that fits the data and explain what the y-intercept means.
Non-proportional; y = 6x + 6. The y-intercept of 6 represents the starting/base fee.
One point is NOT a solution to y = (1/2)x + 2: (0, 2), (2, 3), (4, 4), or (6, 6). Identify the point and justify your answer by substitution.
(6, 6). When x = 6, y = (1/2)(6) + 2 = 5, not 6.
Create a relation with four ordered pairs that IS a function even though at least two ordered pairs have the same y-value. Then explain why your relation is still a function.
Answers vary. Example: {(1, 5), (2, 5), (3, 7), (4, 9)}. It is a function because each x-value has exactly one output.
A table has points (0, 2), (1, 5), (2, 8), and (3, 11). Determine whether the function is linear. Explain using the rate of change, and write an equation for the function.
The function is linear. The y-values increase by 3 each time x increases by 1, so the rate of change is 3. The equation is y = 3x + 2.
Solve the absolute value equation:
|8 - 2x| - 4 = 6
x = -1 or x = 9