Lynn Marecek

Review Exercises

Graph Linear Equations in Two Variables

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system.

391.

ⓐ $(−1, −5)$
ⓑ $(−3, 4)$
ⓒ $(2, −3)$
ⓓ $(1, \frac{5}{2})$

392.

ⓐ $(−2, 0)$
ⓑ $(0, −4)$
ⓒ $(0, 5)$
ⓓ $(3, 0)$

In the following exercises, determine which ordered pairs are solutions to the given equations.

393.

$5 x + y = 10;$
ⓐ $(5, 1)$
ⓑ $(2, 0)$
ⓒ $(4, −10)$

394.

$y = 6 x - 2;$
ⓐ $(1, 4)$
ⓑ $(\frac{1}{3}, 0)$
ⓒ $(6, −2)$

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

395.

$y = 4 x - 3$

396.

$y = −3 x$

397.

$y = \frac{1}{2} x + 3$

398.

$y = - \frac{4}{5} x - 1$

399.

$x - y = 6$

400.

$2 x + y = 7$

401.

$3 x - 2 y = 6$

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

402.

$y = −2$

403.

$x = 3$

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

404.

$y = −2 x$ and $y = −2$

405.

$y = \frac{4}{3} x$ and $y = \frac{4}{3}$

Find x- and y-Intercepts

In the following exercises, find the x- and y-intercepts.

406.

The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 2), (0, 4), (2, 6), and (4, 8).

407.

In the following exercises, find the intercepts of each equation.

408.

$x - y = −1$

409.

$x + 2 y = 6$

410.

$2 x + 3 y = 12$

411.

$y = \frac{3}{4} x - 12$

412.

$y = 3 x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

413.

$- x + 3 y = 3$

414.

$x - y = 4$

415.

$2 x - y = 5$

416.

$2 x - 4 y = 8$

417.

$y = 4 x$

Slope of a Line

Find the Slope of a Line

In the following exercises, find the slope of each line shown.

418.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, 0) and (1, negative 3).

419.

420.

421.

In the following exercises, find the slope of each line.

422.

$y = 2$

423.

$x = 5$

424.

$x = −3$

425.

$y = −1$

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

426.

$(−1, −1), (0, 5)$

427.

$(3.5), (4, −1)$

428.

$(−5, −2), (3, 2)$

429.

$(2, 1), (4, 6)$

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

430.

$(2, −2);$ $m = \frac{5}{2}$

431.

$(−3, 4);$ $m = - \frac{1}{3}$

432.

x-intercept $−4;$ $m = 3$

433.

y-intercept 1; $m = - \frac{3}{4}$

Graph a Line Using Its Slope and Intercept

In the following exercises, identify the slope and y-intercept of each line.

434.

$y = −4 x + 9$

435.

$y = \frac{5}{3} x - 6$

436.

$5 x + y = 10$

437.

$4 x - 5 y = 8$

In the following exercises, graph the line of each equation using its slope and y-intercept.

438.

$y = 2 x + 3$

439.

$y = − x - 1$

440.

$y = - \frac{2}{5} x + 3$

441.

$4 x - 3 y = 12$

In the following exercises, determine the most convenient method to graph each line.

442.

$x = 5$

443.

$y = −3$

444.

$2 x + y = 5$

445.

$x - y = 2$

446.

$y = \frac{2}{2} x + 2$

447.

$y = \frac{3}{4} x - 1$

Graph and Interpret Applications of Slope-Intercept

448.

Katherine is a private chef. The equation $C = 6.5 m + 42$ models the relation between her weekly cost, C, in dollars and the number of meals, m, that she serves.

ⓐ Find Katherine’s cost for a week when she serves no meals.
ⓑ Find the cost for a week when she serves 14 meals.
ⓒ Interpret the slope and C-intercept of the equation.
ⓓ Graph the equation.

449.

Marjorie teaches piano. The equation $P = 35 h - 250$ models the relation between her weekly profit, P, in dollars and the number of student lessons, s, that she teaches.

ⓐ Find Marjorie’s profit for a week when she teaches no student lessons.
ⓑ Find the profit for a week when she teaches 20 student lessons.
ⓒ Interpret the slope and P-intercept of the equation.
ⓓ Graph the equation.

Use Slopes to Identify Parallel and Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

450.

$4 x - 3 y = −1; y = \frac{4}{3} x - 3$

451.

$y = 5 x - 1; 10 x + 2 y = 0$

452.

$3 x - 2 y = 5; 2 x + 3 y = 6$

453.

$2 x - y = 8; x - 2 y = 4$

Find the Equation of a Line

Find an Equation of the Line Given the Slope and y-Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

454.

slope $\frac{1}{3}$ and $y$ -intercept $(0, −6)$

455.

slope $−5$ and $y$ -intercept $(0, −3)$

456.

slope 0 and $y$ -intercept $(0, 4)$

457.

slope $−2$ and $y$ -intercept $(0, 0)$

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

458.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 1), (1, 3), and (2, 5).

459.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 5), (1, 2), and (2, negative 1).

460.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (4, 1), and (8, 4).

461.

This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4).

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

462.

$m = - \frac{1}{4},$ point $(−8, 3)$

463.

$m = \frac{3}{5},$ point $(10, 6)$

464.

Horizontal line containing $(−2, 7)$

465.

$m = −2,$ point $(−1, −3)$

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

466.

$(2, 10)$ and $(−2, −2)$

467.

$(7, 1)$ and $(5, 0)$

468.

$(3, 8)$ and $(3, −4)$

469.

$(5, 2)$ and $(−1, 2)$

Find an Equation of a Line Parallel to a Given Line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

470.

line $y = −3 x + 6,$ point $(1, −5)$

471.

line $2 x + 5 y = −10,$ point $(10, 4)$

472.

line $x = 4,$ point $(−2, −1)$

473.

line $y = −5,$ point $(−4, 3)$

Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

474.

line $y = - \frac{4}{5} x + 2,$ point $(8, 9)$

475.

line $2 x - 3 y = 9,$ point $(−4, 0)$

476.

line $y = 3,$ point $(−1, −3)$

477.

line $x = −5$ point $(2, 1)$

Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

478.

Determine whether each ordered pair is a solution to the inequality $y < x - 3 :$

ⓐ $(0, 1)$ ⓑ $(−2, −4)$ ⓒ $(5, 2)$ ⓓ $(3, −1)$
ⓔ $(−1, −5)$

479.

Determine whether each ordered pair is a solution to the inequality $x + y > 4 :$

ⓐ $(6, 1)$ ⓑ $(−3, 6)$ ⓒ $(3, 2)$ ⓓ $(−5, 10)$ ⓔ $(0, 0)$

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

480.

Write the inequality shown by the graph with the boundary line $y = − x + 2 .$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 2), (1, 1), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.

481.

Write the inequality shown by the graph with the boundary line $y = \frac{2}{3} x - 3 .$

482.

Write the inequality shown by the shaded region in the graph with the boundary line $x + y = −4 .$

483.

Write the inequality shown by the shaded region in the graph with the boundary line $x - 2 y = 6 .$

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

484.

Graph the linear inequality $y > \frac{2}{5} x - 4 .$

485.

Graph the linear inequality $y \leq - \frac{1}{4} x + 3 .$

486.

Graph the linear inequality $x - y \leq 5 .$

487.

Graph the linear inequality $3 x + 2 y > 10 .$

488.

Graph the linear inequality $y \leq −3 x .$

489.

Graph the linear inequality $y < 6 .$

Solve Applications using Linear Inequalities in Two Variables

490.

Shanthie needs to earn at least $500 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays $10 an hour and the other as an intern in a law office for $25 hour. How many hours does Shanthie need to work at each job to earn at least $500 per week?

ⓐ Let x be the number of hours she works teaching swimming and let y be the number of hours she works as an intern. Write an inequality that would model this situation.
ⓑ Graph the inequality.
ⓒ Find three ordered pairs $(x, y)$ that would be solutions to the inequality. Then, explain what that means for Shanthie.

491.

Atsushi he needs to exercise enough to burn 600 calories each day. He prefers to either run or bike and burns 20 calories per minute while running and 15 calories a minute while biking.

ⓐ If x is the number of minutes that Atsushi runs and y is the number minutes he bikes, find the inequality that models the situation.
ⓑ Graph the inequality.
ⓒ List three solutions to the inequality. What options do the solutions provide Atsushi?

Relations and Functions

Find the Domain and Range of a Relation

In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation.

492.

${(5, −2), (5, −4), (7, −6),$
$(8, −8), (9, −10)}$

493.

${(−3, 7), (−2, 3), (−1, 9),$
$(0, −3), (−1, 8)}$

In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

494.

The mapping below shows the average weight of a child according to age.

This figure shows two table that each have one column. The table on the left has the header “Age (yrs)” and lists the numbers 1, 2, 3, 4, 5, 6, and 7. The table on the right has the header “Weight (pounds)” and lists the numbers 20, 35, 30, 45, 40, 25, and 50. There are arrows starting at numbers in the age table and pointing towards numbers in the weight table. The first arrow goes from 1 to 20. The second arrow goes from 2 to 25. The third arrow goes from 3 to 30. The fourth arrow goes from 4 to 35. The fifth arrow goes from 5 to 40. The sixth arrow goes from 6 to 45. The seventh arrow goes from 7 to 50.

In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

495.

The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 1), (negative 2, negative 1), (negative 2, negative 3), (0, negative 1), (0, 4), and (4, 3).

Determine if a Relation is a Function

In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.

496.

${(9, −5), (4, −3), (1, −1),$
$(0, 0), (1, 1), (4, 3), (9, 5)}$

497.

${(−3, 27), (−2, 8), (−1, 1),$
$(0, 0), (1, 1), (2, 8), (3, 27)}$

In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function.

498.

This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fourth power” and lists the numbers 0, 1, 16, and 81. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fourth power table. The first arrow goes from negative 3 to 81. The second arrow goes from negative 2 to 16. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 16. The seventh arrow goes from 3 to 81.

499.

In the following exercises, determine whether each equation is a function.

500.

$2 x + y = −3$

501.

$y = x^{2}$

502.

$y = 3 x - 5$

503.

$y = x^{3}$

504.

$2 x + y^{2} = 4$

Find the Value of a Function

In the following exercises, evaluate the function:

505.

$f (x) = 3 x - 4$

506.

$f (x) = −2 x + 5$

507.

$f (x) = x^{2} - 5 x + 6$

508.

$f (x) = 3 x^{2} - 2 x + 1$

In the following exercises, evaluate the function.

509.

$g (x) = 3 x^{2} - 5 x;$ $g (2)$

510.

$F (x) = 2 x^{2} - 3 x + 1;$
$F (−1)$

511.

$h (t) = 4 | t - 1 | + 2;$ $h (−3)$

512.

$f (x) = \frac{x + 2}{x - 1};$ $f (3)$

Graphs of Functions

Use the Vertical line Test

In the following exercises, determine whether each graph is the graph of a function.

513.

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).

514.

The figure has an s-shaped function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curve goes through the points (negative 1, negative 1), (0, 0), and (1, 1).

515.

The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 5, 0), (5, 0), (0, negative 5), and (0, 5).

516.

The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (2, 2), and (2, negative 2). The left-most point on the graph is (negative 2, 0).

517.

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).

518.

The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 3, 0), (negative 4, 2), and (negative 4, negative 2). The curved line on the right goes through the points (3, 0), (4, 2), and (4, negative 2).

519.

The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, negative 1) and goes to the right. The line goes through the points (1, 0), (1, negative 2), (2, 1), and (2, negative 3).

Identify Graphs of Basic Functions

In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.

520.

$f (x) = 5 x + 1$

521.

$f (x) = −4 x - 2$

522.

$f (x) = \frac{2}{3} x - 1$

523.

$f (x) = −6$

524.

$f (x) = 2 x$

525.

$f (x) = 3 x^{2}$

526.

$f (x) = - \frac{1}{2} x^{2}$

527.

$f (x) = x^{2} + 2$

528.

$f (x) = x^{3} - 2$

529.

$f (x) = \sqrt{x + 2}$

530.

$f (x) = − | x |$

531.

$f (x) = | x | + 1$

Read Information from a Graph of a Function

In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation

532.

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).

533.

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 2). The line goes through the points (negative 1, 3) and (1, 3).

534.

The figure has a cubic function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 4), (0, 0), and (2, 4).

In the following exercises, use the graph of the function to find the indicated values.

535.

This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, 1), (negative pi, 0), (negative 1 divided by 2 times pi, negative 1), (0, 0), (1 divided by 2 times pi, 1), (pi, 0), (3 divided by 2 times pi, negative 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, 1) and (1 divided by 2 times pi, 1) are the highest points on the graph. The points (negative 1 divided by 2 times pi, negative 1) and (3 divided by 2 times pi, negative 1) are the lowest points on the graph. The pattern extends infinitely to the left and right.

ⓐ Find $f (0) .$
ⓑ Find $f (\frac{1}{2} π) .$
ⓒ Find $f (- \frac{3}{2} π) .$
ⓓ Find the values for x when $f (x) = 0 .$
ⓔ Find the $x$ -intercepts.
ⓕ Find the $y$ -intercepts.
ⓖ Find the domain. Write it in interval notation.
ⓗ Find the range. Write it in interval notation.

536.

The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0). The point (0, 2) is the highest point on the graph.

ⓐ Find $f (0) .$
ⓑ Find the values for x when $f (x) = 0 .$
ⓒ Find the $x$ -intercepts.
ⓓ Find the $y$ -intercepts.
ⓔ Find the domain. Write it in interval notation.
ⓕ Find the range. Write it in interval notation.