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Math  /  Algebra

QuestionThe graph of a function is shown on the coordinate plane below.
Which relationship represents a function with a lesser rate of change than the function graphed?
A \begin{tabular}{|c|c|} \hlinexx & yy \\ \hline-4 & -9 \\ \hline 0 & -4 \\ \hline 4 & 1 \\ \hline 8 & 6 \\ \hline \end{tabular}
C \begin{tabular}{|c|c|} \hlinexx & yy \\ \hline 3 & -14 \\ \hline 6 & -26 \\ \hline 9 & -38 \\ \hline 12 & -50 \\ \hline \end{tabular}
B y=−6x−3y=-6 x-3
D y=3x+2y=3 x+2

Studdy Solution

STEP 1

1. The graph of the function is a straight line, indicating a linear function.
2. The rate of change of a linear function is its slope.
3. We need to compare the slope of the given graph with the slopes of the other relationships to find one with a lesser rate of change.

STEP 2

1. Calculate the slope of the given graph.
2. Determine the slopes of the other relationships.
3. Compare the slopes to identify the relationship with a lesser rate of change.

STEP 3

Calculate the slope of the given graph using the points (0, 0) and (3, -6).
The formula for slope m m is: m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1}
Substitute the points (0, 0) and (3, -6) into the formula: m=−6−03−0=−63=−2 m = \frac{-6 - 0}{3 - 0} = \frac{-6}{3} = -2
The slope of the given graph is −2-2.

STEP 4

Calculate the slope for each of the relationships:
- **Relationship A:** Use points (-4, -9) and (8, 6). m=6−(−9)8−(−4)=1512=54 m = \frac{6 - (-9)}{8 - (-4)} = \frac{15}{12} = \frac{5}{4}
- **Relationship B:** The equation is y=−6x−3 y = -6x - 3 . The slope is −6-6.
- **Relationship C:** Use points (3, -14) and (6, -26). m=−26−(−14)6−3=−123=−4 m = \frac{-26 - (-14)}{6 - 3} = \frac{-12}{3} = -4
- **Relationship D:** The equation is y=3x+2 y = 3x + 2 . The slope is 33.

STEP 5

Compare the slopes: - Given graph slope: −2-2 - Relationship A slope: 54\frac{5}{4} - Relationship B slope: −6-6 - Relationship C slope: −4-4 - Relationship D slope: 33
The relationship with a lesser rate of change than −2-2 is Relationship A with a slope of 54\frac{5}{4}.
The relationship with a lesser rate of change is:
A \boxed{A}

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