Question

part 1 of 4
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

a. adding k to the outputs of f will result in which of the following?
a. the graph will be shifted to the right.
b. the graph will be shifted to the left.
c. the graph will be shifted downward.
d. the graph will be shifted upward.
e. the graph will have a negative slope, and a y-intercept will change signs.
f. the graph will have a negative slope; there will be no change in the y-intercept.
g. the graph will have a positive slope, and a y-intercept will change signs.
h. the graph will have a positive slope; there will be no change in the y-intercept.

Explanation:

Step1: Recall Vertical Shift Rule

For a function \( y = f(x) \), adding a constant \( k \) to the output (i.e., \( y = f(x)+k \)) causes a vertical shift. If \( k>0 \), it's a shift upward; if \( k < 0 \), it's a shift downward.

Step2: Analyze the Given \( k < 0 \)

Here, we are adding \( k \) (where \( k < 0 \)) to the outputs of \( f \). So the transformation is \( y = f(x)+k \), and since \( k \) is negative, this is equivalent to a vertical shift downward. The slope of a linear function is determined by the rate of change of \( y \) with respect to \( x \); adding a constant to the output does not change the slope (it only affects the \( y \)-intercept's position vertically). So the graph will be shifted downward.

Answer:

C. The graph will be shifted downward.