Writing Linear Equations and Scatter Plots: math practice, Grade 9

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Math practice Grade 9 : Writing Linear Equations and Scatter Plots — Zyro the alien explorer of Planète Maths

22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Slope-intercept form ★★★

(a) Write the equation of the line with slope $4$ and y-intercept $-3$.

(b) Write the equation of the line with slope $-\dfrac{1}{2}$ and y-intercept $6$.

2 Point-slope practice ★★★

Write each line in slope-intercept form.

(a) Slope $2$, through $(1, 4)$.

(b) Slope $-3$, through $(-2, 5)$.

3 Two easy points ★★★

A line passes through $(1, 2)$ and $(5, 10)$. Find its slope and its equation.

4 Name the correlation ★★★

Say whether each pair of variables most likely has a positive correlation, a negative correlation, or no correlation.

(a) The outside temperature and the cups of hot chocolate sold at a stand.

(b) A person’s height and shoe size.

5 Find a residual ★★★

A model for the number of pages read is $y = 3x + 10$, where $x$ is the number of minutes. Find the residual at each data point.

(a) $(4, 25)$ (b) $(6, 26)$

6 True or false? ★★★

Decide whether each statement is true or false, and explain.

(a) A residual of 0 means the data point lies exactly on the line.

(b) If two variables have a strong correlation, one must cause the other.

7 Evaluate piecewise and absolute value functions ★★★

Let $f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ 3x & \text{if } x \ge 0 \end{cases}$ and $g(x) = |x - 3|$.

Find $f(-4)$, $f(0)$, $f(5)$, $g(-1)$, $g(3)$, and $g(7)$.

8 Filling a pool ★★★

A small pool already holds some water. After 2 minutes it holds 30 gallons, and the hose adds water at a steady 12 gallons per minute (about 45 liters per minute).

(a) Write an equation for the volume $V$ after $t$ minutes.

(b) How many gallons are in the pool after 10 minutes?

9 Taxi fare ★★★

A 3-mile ride costs $9.50 and an 8-mile ride costs $19.50 (1 mile is about 1.6 km).

(a) Write the fare $F$ as a linear function of the miles $x$.

(b) What is the starting fee, and what does the slope mean?

10 Residual table ★★★

A line of best fit for some data is $y = 2x + 1$. Complete the table.

$x$	Actual $y$	Predicted	Residual
2	4
4	10
6	12
8	19

Do the residuals suggest that the line is a reasonable model?

11 Reading r ★★★

Three scatter plots have $r = -0.92$, $r = 0.15$, and $r = 0.97$.

(a) Describe the direction and strength of each.

(b) For which two would a linear model be most appropriate?

12 Bike rental model ★★★

A bike rental shop uses $y = 0.4x + 25$, where $x$ is the miles ridden and $y$ is the cost in dollars.

(a) Interpret the slope and the y-intercept.

(b) Predict the cost for 120 miles.

13 Interpolation or extrapolation ★★★

The model $y = 4.5x + 51$ was built from quiz data with study times from 1 to 8 hours.

(a) Predict the score for 4.5 hours. Is this interpolation or extrapolation?

(b) Predict the score for 12 hours. Is this reliable? Explain.

14 Parking fees ★★★

A garage charges $4 for any stay of up to 2 hours, and then $3 for each additional hour. So $P(h) = 4$ for $0 < h \le 2$ and $P(h) = 4 + 3(h - 2)$ for $h > 2$.

(a) Find $P(1.5)$ and $P(5)$.

(b) Write the second rule in the form $3h + c$.

15 A negative slope ★★★

A line passes through $(2, -1)$ and $(-4, 8)$.

(a) Write its equation in slope-intercept form.

(b) Find its x-intercept.

16 Standard form from intercepts ★★★

A line crosses the x-axis at $(6, 0)$ and the y-axis at $(0, -4)$. Write its equation in slope-intercept form and in standard form $Ax + By = C$ with integer coefficients.

17 Missing coordinate ★★★

The points $(2, k)$ and $(6, 13)$ lie on a line of slope $\dfrac{3}{2}$. Find $k$ and write the equation of the line.

18 A pattern in the residuals ★★★

A drone’s height $y$ (in feet) was recorded at $x$ seconds: $(1, 10)$, $(2, 6)$, $(3, 4)$, $(4, 4)$, $(5, 6)$, $(6, 10)$. A student proposes the model $y = 7$.

(a) Compute the six residuals.

(b) Describe the residual pattern. Is a line a good model for these data?

19 Fit a line through two data points ★★★

An app had these downloads (in thousands) after $x$ weeks: $(0, 8)$, $(2, 15)$, $(4, 18)$, $(6, 27)$, $(8, 32)$.

(a) Write the equation of the line through the first and last points.

(b) Compute the residual at each of the five points.

20 Hidden cause ★★★

A town finds that the more ice cream cones it sells each week, the more sunburns the clinic treats ($r = 0.9$).

(a) Does eating ice cream cause sunburns?

(b) Name a lurking variable and explain how it creates the correlation.

21 Equation of a V graph ★★★

The graph below is the graph of $y = a|x - h| + k$. Find $a$, $h$, and $k$, then the x-intercepts.

22 Write a piecewise function ★★★

A function is made of two line segments. For $x \le 4$ the line passes through $(0, 2)$ and $(4, 6)$. For $x > 4$ the line passes through $(4, 6)$ and $(8, 0)$.

(a) Write both rules.

(b) Evaluate the function at $x = 2$ and $x = 6$.