Questions: The graph of a linear function f has a negative slope. Describe the effect on the graph of the function if the transformation has a value of k<0. a. adding k to the outputs of f b. adding k to the inputs of f c. multiplying the outputs of f by k d. multiplying the inputs of f by k c. Multiplying the outputs of f by k will result in which of the following? A. The graph will have a positive slope, and a y-intercept will change signs. B. The graph will have a negative slope; there will be no change in the y-intercept. C. The graph will be shifted downward. D. The graph will be shifted to the right. E. The graph will be shifted upward. F. The graph will be shifted to the left. G. The graph will have a positive slope; there will be no change in the y-intercept. H. The graph will have a negative slope, and a y-intercept will change signs.

The graph of a linear function f has a negative slope. Describe the effect on the graph of the function if the transformation has a value of k<0.
a. adding k to the outputs of f
b. adding k to the inputs of f
c. multiplying the outputs of f by k
d. multiplying the inputs of f by k
c. Multiplying the outputs of f by k will result in which of the following?
A. The graph will have a positive slope, and a y-intercept will change signs.
B. The graph will have a negative slope; there will be no change in the y-intercept.
C. The graph will be shifted downward.
D. The graph will be shifted to the right.
E. The graph will be shifted upward.
F. The graph will be shifted to the left.
G. The graph will have a positive slope; there will be no change in the y-intercept.
H. The graph will have a negative slope, and a y-intercept will change signs.

Transcript text: The graph of a linear function $f$ has a negative slope. Describe the effect on the graph of the function if the transformation has a value of $\mathrm{k}<0$. a. adding $k$ to the outputs of $f$ b. adding k to the inputs of $f$ c. multiplying the outputs of $f$ by $k$ d. multiplying the inputs of $f$ by $k$ c. Multiplying the outputs of $f$ by $k$ will result in which of the following? A. The graph will have a positive slope, and a $y$-intercept will change signs. B. The graph will have a negative slope; there will be no change in the $y$-intercept. C. The graph will be shifted downward. D. The graph will be shifted to the right. E. The graph will be shifted upward. F. The graph will be shifted to the left. G. The graph will have a positive slope; there will be no change in the $y$-intercept. H. The graph will have a negative slope, and a y-intercept will change signs.

Solution

Solution Steps

Step 1: Adding $ k $ to the outputs of $ f $

The transformation $ f(x) + k $ shifts the graph of $ f $ vertically.
Since $ k < 0 $, the graph will shift downward by $ |k| $ units.
The slope of the graph remains unchanged because the transformation only affects the $ y $-intercept.

Step 2: Adding $ k $ to the inputs of $ f $

The transformation $ f(x + k) $ shifts the graph of $ f $ horizontally.
Since $ k < 0 $, the graph will shift to the right by $ |k| $ units.
The slope of the graph remains unchanged because the transformation only affects the horizontal position.

Step 3: Multiplying the outputs of $ f $ by $ k $

The transformation $ k \cdot f(x) $ scales the graph of $ f $ vertically.
Since $ k < 0 $, the graph will be reflected over the $ x $-axis and scaled by $ |k| $.
The slope of the graph will change sign (from negative to positive) because of the reflection.
The $ y $-intercept will also change sign due to the reflection.

Final Answer

a. The graph will be shifted downward.
b. The graph will be shifted to the right.
c. The graph will have a positive slope, and a $ y $-intercept will change signs.
d. The graph will have a negative slope; there will be no change in the $ y $-intercept.

The correct answer is C.

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