Question

d. multiplying the inputs of f by k will result in which of the following?
a. the graph will be shifted upward.

b. the graph will be shifted to the right.

c. the graph will have a negative slope, and a y-intercept will change signs.

d. the graph will be shifted downward.

e. the graph will be shifted to the left.

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:

To determine the effect of multiplying the inputs of a function \( f \) by \( k \), we analyze function transformations. When we have a function \( y = f(x) \), multiplying the input \( x \) by \( k \) gives \( y = f(kx) \). If \( k>0 \), this is a horizontal scaling and shift. For a linear function (assuming \( f(x) \) is linear, e.g., \( f(x)=mx + b \)), multiplying inputs by \( k \) (scaling \( x \) by \( k \)) changes the slope (if \( k
eq1 \)) but the key here is about horizontal shifts. However, the options relate to slope and y - intercept or shifts. Wait, maybe there's a mis - understanding. Wait, if we consider the original function (maybe linear) and multiplying inputs (x - values) by k. But actually, when we have a function transformation, multiplying the input by k (for \( k > 0\)) is a horizontal compression (if \( k>1\)) or stretch (if \( 0 < k < 1\)). But the options given are about slope and y - intercept or shifts. Wait, maybe the original function is linear, say \( f(x)=mx + b \). If we multiply the input (x) by k, we get \( f(kx)=m(kx)+b = mkx + b \). The slope is now \( mk \), and the y - intercept is still \( b \) (no change). But the options about slope: if we assume that maybe there was a mis - statement, and the actual transformation is about multiplying the function by k, but the question says "multiplying the inputs of f by k". Wait, maybe the intended function is linear, and when we multiply the input (x) by k, for the purpose of the question, if we consider the effect on slope and y - intercept. But the correct option here, looking at the options, option H says "The graph will have a positive slope, there will be no change in the y - intercept." Wait, no, maybe I made a mistake. Wait, actually, when you multiply the input (x) of a linear function \( y = mx + b \) by k, you get \( y=m(kx)+b=mkx + b \). The y - intercept (the value when x = 0) is still \( b \) (since when x = 0, \( y = b \)), so the y - intercept doesn't change. The slope becomes \( mk \). If the original slope \( m \) was positive, and k is positive (since we are multiplying inputs, k is likely positive), then the new slope \( mk \) is also positive. So the graph will have a positive slope, and there will be no change in the y - intercept, which is option H.

Answer:

H. The graph will have a positive slope, there will be no change in the y - intercept.