Question

what is the value of d in the function $f(x) = \cos(x) + d$ shown in the graph? (1 point) the value of d in the function $f(x) = \cos(x) + d$ is $\square$.

Explanation:

Step1: Recall the standard cosine function

The standard function is \( y = \cos(x) \), which has a maximum value of \( 1 \) and a minimum value of \( -1 \), and its midline (vertical shift) is \( y = 0 \).

Step2: Analyze the given graph's midline

For the function \( f(x)=\cos(x)+d \), the vertical shift \( d \) is the midline of the graph. Looking at the graph, the maximum value is \( -2 \) and the minimum value is \( -4 \). The midline (average of max and min) is \( \frac{-2 + (-4)}{2}=\frac{-6}{2}=-3 \). Alternatively, we can look at the y - intercept. When \( x = 0 \), \( f(0)=\cos(0)+d=1 + d \). From the graph, when \( x = 0 \), \( f(0)=-2 \). So we set up the equation \( 1 + d=-2 \).

Step3: Solve for \( d \)

From \( 1 + d=-2 \), subtract \( 1 \) from both sides: \( d=-2 - 1=-3 \).

Answer:

\( -3 \)