Exercises: Understand graphs of equations

A

Warm-Up: Review What You Know

These problems review skills you already have.

1.

Evaluate $y = 3x - 1$ when $x = 4$ .

A.

11

B.

13

C.

9

D.

7

2.

Which statement correctly describes the solution to the one-variable equation $x = 5$ ?

A.

The solution is the single point $x = 5$ on a number line.

B.

The solution is the line $y = 5$ in the coordinate plane.

C.

The solution is every ordered pair $(5, y)$ for any value of $y$ .

D.

The solution is every ordered pair $(x, 5)$ for any value of $x$ .

3.

A student says the ordered pair $(2, 7)$ satisfies the equation $y = 3x + 1$ .
Is the student correct?

A.

Yes, because $3(2) + 1 = 7$ .

B.

No, because $2 + 7 \neq 1$ .

C.

Yes, because $2 + 7 = 9$ and $3 + 1 = 4$ .

D.

No, because $3(7) + 1 \neq 2$ .

B

Fluency Practice

Determine whether each given point lies on the graph of the equation. Show your substitution.

C

Varied Practice

These problems use a variety of formats and representations.

1.

Which of the following best describes what the graph of a two-variable equation represents?

A.

A sample of some solutions chosen to make the graph look nice.

B.

The set of all ordered pairs $(x, y)$ that make the equation true.

C.

Only the points where the equation crosses an axis.

D.

The solutions to the equation when $x > 0$ .

2.

To verify that $(-1, -5)$ lies on the graph of $y = 3x - 2$ , substitute $x = -1$ into the right side:
$3(\underline{\hspace{5em}}) - 2 = \underline{\hspace{5em}}$ . Since this equals $y = \underline{\hspace{5em}}$ , the point ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ on the graph.

x-value substituted:

computed value:

y-coordinate:

is or is not:

3.

A student graphs $y = x^2 - 4$ and plots only the points $(-2, 0)$ , $(0, -4)$ , and $(2, 0)$ .
She then says "those three points are the complete graph." What is wrong with her claim?

A.

She made arithmetic errors — those points are not on the graph.

B.

The graph of $y = x^2 - 4$ is a line, not a collection of points.

C.

Those three points are on the graph, but the graph also includes infinitely many other points between and beyond them.

D.

The graph only exists for $x \geq 0$ , so she cannot plot $(-2, 0)$ .

$A blank table of values for y = -x + 3 with x-values -1, 0, 2, and 4, and empty y-value cells.$

4.

Complete the table of values for $y = -x + 3$ , then identify the shape of its graph.

$x$	$y = -x + 3$
$-1$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$0$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$2$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
$4$	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

The graph of this equation is a ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .

y when x = -1:

y when x = 0:

y when x = 2:

y when x = 4:

graph shape:

5.

The point $(k, -3)$ lies on the graph of $y = 2x - 7$ . What is the value of $k$ ?

D

Word Problems

Read each scenario carefully. Use substitution or graphing to answer.

1.

A candle's height $h$ (in inches) after $t$ hours of burning is modeled by the equation
$h = 8 - 1.5t$ .

Is the point $(4, 2)$ on the graph of this equation? What does the point represent in context?
First, find the value of $h$ when $t = 4$ (enter a number).

$A graph of height versus time for a thrown ball, showing a downward parabola with vertex at (2, 64) and x-intercepts at 0 and 4 seconds.$

2.

A ball is thrown upward and its height $y$ (in feet) after $x$ seconds is modeled by
$y = -16x^2 + 64x$ .

1.

What is the height of the ball at $x = 1$ second? Is the point $(1, 48)$ on the graph?

A.

Height = 48 ft; yes, $(1, 48)$ is on the graph.

B.

Height = 48 ft; no, $(1, 48)$ is not on the graph because the ball is still rising.

C.

Height = 80 ft; yes, $(1, 80)$ is on the graph.

D.

Height = 32 ft; no, the ball hasn't reached its peak yet so we can't read the graph.

2.

At what time $x$ (in seconds) does the ball land (return to height $y = 0$ )?
Use the graph or solve algebraically.

3.

A store's daily profit $P$ (in dollars) from selling $x$ items is modeled by $P = 5x - 20$ .

How many items must be sold for the store to break even (profit = 0)? That is, find the
$x$ -intercept of the graph of $P = 5x - 20$ .

4.

The graph of $y = x^2 - 4x$ passes through the points $(0, 0)$ and $(4, 0)$ .

What do these two $x$ -intercepts tell you about the solutions to the equation
$x^2 - 4x = 0$ ? Explain using the definition of what a graph represents.

E

Error Analysis

Each problem shows a student's reasoning. Identify the error.

1.

Marcus says: "The line $y = 2x + 1$ only has three solutions:
$(0, 1)$ , $(1, 3)$ , and $(2, 5)$ . Those are the only points I can see on my graph paper,
so those are the only solutions."

What is the fundamental error in Marcus's reasoning?

A.

Marcus computed the $y$ -values incorrectly.

B.

Marcus confused the graph of a two-variable equation with a one-variable equation.

C.

Marcus thinks the graph ends at the edge of his paper, but the graph extends infinitely and has infinitely many solutions.

D.

Marcus should have plotted only integer values of $x$ .

2.

Priya is graphing $y = x^2 - 1$ . She builds a table:

| $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ |
| $y$ | $3$ | $0$ | $-1$ | $0$ | $3$ |

She then says: "The graph of $y = x^2 - 1$ consists of exactly these 5 points."

What does Priya need to understand to correct her mistake?

A.

She computed some $y$ -values incorrectly.

B.

She plotted points in the wrong quadrant.

C.

A table gives a discrete sample of solutions. The actual graph is the smooth parabola through all these points — infinitely many solutions, not just the five she computed.

D.

She should only plot points where $y \geq 0$ , since graphs only show positive values.

F

Challenge / Extension

These bonus problems require multi-step thinking.

1.

The graph of $x^2 + y^2 = 100$ is a circle. Three of the following four points lie on
this circle: $(6, 8)$ , $(8, 6)$ , $(10, 0)$ , and $(5, 9)$ .
What is the $y$ -coordinate of the point that does NOT lie on the circle?
(Enter the $y$ -coordinate of the non-solution point.)