Question

lesson 10: practice problems

1. the graph shows the distance of a car from home as a function of time.

describe what a person watching the car may be seeing.

1. the equation and the graph represent two functions. use the equation ( y = 4 ) and the graph to answer the questions.

a. when ( x ) is 4, is the output of the equation or the graph greater?
b. what value for ( x ) produces the same output in both the graph and the equation?

Explanation:

Problem 1

1. First segment (rising line): The car moves away from home at a constant speed, so its distance from home increases steadily over time.
2. Second segment (flat line): The car stops moving and stays at a fixed distance from home for a period of time.
3. Third segment (falling line): The car travels back towards home at a constant speed, so its distance from home decreases steadily until it reaches home (distance = 0).

First, find the equation of the graphed line. The line has a y-intercept of 1, and slope $m=\frac{1-0}{2-(-2)}=\frac{1}{4}$. So its equation is $y=\frac{1}{4}x + 1$.

Part a

Step1: Find equation output at x=4

For $y=4$, when $x=4$, output is $4$.

Step2: Find graph output at x=4

Substitute $x=4$ into $y=\frac{1}{4}x + 1$:
$\frac{1}{4}(4) + 1 = 1 + 1 = 2$

Step3: Compare the two outputs

$4 > 2$, so the equation's output is greater.

Part b

Step1: Set functions equal

Set $\frac{1}{4}x + 1 = 4$

Step2: Solve for x

Subtract 1 from both sides: $\frac{1}{4}x = 3$
Multiply by 4: $x = 3 \times 4 = 12$

Answer:

First, the car drives away from home at a constant speed, getting farther over time. Then, the car stops and stays in one place (not moving closer or farther from home) for some time. Finally, the car drives back home at a constant speed, getting closer until it arrives back home.

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Problem 2