Slope and Graphing Linear Functions: math practice, Grade 9 – download the PDF

1 Shrinking candle ★★★

A candle is 18 cm tall after burning for 1 hour and 9 cm tall after burning for 4 hours.

1. Find the rate of change of its height.
2. What does the sign of the rate tell you?
3. At this rate, how tall is the candle after 6 hours?

2 Four slopes ★★★

Find the slope of the line through each pair of points, or write “undefined”.

1. \((2,\,3)\) and \((6,\,11)\)
2. \((-3,\,4)\) and \((5,\,-12)\)
3. \((7,\,-1)\) and \((7,\,9)\)
4. \((-5,\,0)\) and \((0,\,0)\)

3 Read the slope and intercept ★★★

Give the slope and the \(y\)-intercept of each line.

1. \(y=5x-7\)
2. \(y=-\dfrac34x+2\)
3. \(y=6-x\)
4. \(y=\dfrac{x}{2}\)

4 Write the equation ★★★

Write the equation in slope-intercept form of the line with:

1. slope \(-2\) and \(y\)-intercept \(9\);
2. slope \(\dfrac13\) and \(y\)-intercept \(-4\);
3. slope \(0\) and \(y\)-intercept \(5\).

5 Find the intercepts ★★★

Find the \(x\)-intercept and the \(y\)-intercept of each line.

1. \(2x+y=8\)
2. \(x-3y=9\)

6 Direct variation or not? ★★★

Which equations show direct variation? When they do, give the constant of variation.

1. \(y=7x\)
2. \(y=7x+1\)
3. \(y=-\dfrac{x}{3}\)
4. \(y=\dfrac4x\)

7 Read a graph ★★★

The line below passes through \(A\), \(B\) and \(C\).

1. Find the slope using \(A\) and \(B\), then check it with \(B\) and \(C\).
2. Give the \(y\)-intercept and write the equation of the line.

8 Into standard form ★★★

Write each equation in standard form \(Ax+By=C\) with integer coefficients, then use \(-\dfrac AB\) to confirm the slope.

1. \(y=\dfrac25x-3\)
2. \(y=-\dfrac34x+6\)

9 Slope and a point ★★★

Use point-slope form to write the equation, in slope-intercept form, of the line with slope \(5\) through \((3,\,-2)\).

10 Through two points ★★★

Find the equation of the line through \((-1,\,6)\) and \((3,\,-2)\).

11 Parallel, perpendicular, or neither? ★★★

Classify each pair of lines.

1. \(y=3x+2\) and \(y=3x-5\)
2. \(y=3x+2\) and \(y=-\dfrac13x+4\)
3. \(2x+3y=6\) and \(3x-2y=8\)
4. \(y=2x\) and \(y=-2x\)

12 Gym membership ★★★

A gym charges a 25-dollar sign-up fee plus 18 dollars per month. Let \(C\) be the total cost in dollars after \(m\) months.

1. Write an equation for \(C\).
2. What do the slope and the \(y\)-intercept mean here?
3. Find the cost after 8 months.
4. After how many months has a member paid exactly 277 dollars in total?

13 Fuel economy ★★★

The distance \(d\) a car travels varies directly with the fuel \(g\) it uses. The car goes 128 miles on 4 gallons.

1. Find the constant of variation and write the equation.
2. How far does the car go on 9 gallons?
3. How many gallons are needed for a 400-mile trip?

14 Graph with intercepts ★★★

Consider the line \(4x+3y=12\).

1. Find both intercepts and describe how to graph the line.
2. Find its slope.
3. Does the point \((6,\,-4)\) lie on the line?

15 Find the missing coordinate ★★★

1. The line through \((k,\,3)\) and \((5,\,9)\) has slope \(2\). Find \(k\).
2. The line through \((4,\,k)\) and \((-2,\,6)\) has slope \(-\dfrac32\). Find \(k\).

16 A perpendicular line ★★★

Find an equation, in standard form, of the line perpendicular to \(3x+4y=8\) that passes through \((6,\,1)\).

17 A parallel line ★★★

Write the equation, in slope-intercept form and in standard form, of the line parallel to \(y=-\dfrac25x+1\) that passes through \((5,\,2)\).

18 A right triangle? ★★★

The vertices of a triangle are \(A(1,\,2)\), \(B(5,\,4)\) and \(C(3,\,8)\).

Use slopes to decide whether the triangle has a right angle, and if so, at which vertex.

19 Taxi fare ★★★

A taxi ride of 4 miles costs 12.50 dollars and a ride of 10 miles costs 26 dollars. The fare is linear in the distance.

1. Find the rate per mile and the starting fee.
2. Write the cost \(C\) in terms of the distance \(m\) in miles.
3. What is the longest whole-number ride you can take with 40 dollars?

20 Fahrenheit and Celsius ★★★

Water freezes at \(0^\circ\text{C}\) (\(32^\circ\text{F}\)) and boils at \(100^\circ\text{C}\) (\(212^\circ\text{F}\)). The relation between the two scales is linear.

1. Find the slope and the equation \(F=mC+b\).
2. Convert \(35^\circ\text{C}\) to Fahrenheit and \(68^\circ\text{F}\) to Celsius.
3. At what temperature do both scales show the same number?
4. Write the equation in standard form.

21 Where two lines meet ★★★

Line \(\ell_1\) has equation \(y=2x-1\). Line \(\ell_2\) is perpendicular to \(\ell_1\) and has \(y\)-intercept \(9\). Find the equation of \(\ell_2\) and the coordinates of the point where the two lines cross.

22 True or false? ★★★

Decide whether each statement is true or false and justify your answer.

1. The line \(y=2x+3\) shows direct variation.
2. If \(y\) varies directly with \(x\), doubling \(x\) doubles \(y\).
3. The line through \((2,\,6)\) and \((5,\,15)\) shows direct variation.
4. A vertical line has slope \(0\).