Question

find the value of $f(7)$. $y = f(x)$

Explanation:

Step1: Identify the graph's peak

The graph of \( y = f(x) \) has a peak at \( x = 6 \) with \( y = 8 \).

Step2: Analyze the slope after \( x = 6 \)

After \( x = 6 \), the graph is decreasing. From \( x = 6 \) (where \( y = 8 \)) to \( x = 7 \), we can observe the pattern or calculate the change. Looking at the graph, the line from \( x = 6 \) ( \( y = 8 \)) to \( x = 8 \) (let's assume the next point) – but for \( x = 7 \), since the slope from \( x = 6 \) to \( x = 8 \) would be \( \frac{y_2 - y_1}{x_2 - x_1} \). Wait, alternatively, notice that before \( x = 6 \), the line was increasing. Let's check the increasing part: from \( x = 0 \) ( \( y = 2 \)) to \( x = 6 \) ( \( y = 8 \) ), the slope is \( \frac{8 - 2}{6 - 0} = 1 \). So the equation of the increasing line is \( y = x + 2 \). Wait, but at \( x = 6 \), \( y = 8 \) (since \( 6 + 2 = 8 \)). Then, after \( x = 6 \), the graph is decreasing. Let's see the next point: when \( x = 8 \), what's \( y \)? Wait, maybe the graph is symmetric? Wait, no. Wait, the graph before \( x = 6 \) is a straight line with slope 1 (from \( x = -2 \) ( \( y = 0 \)) to \( x = 6 \) ( \( y = 8 \)): slope \( \frac{8 - 0}{6 - (-2)} = 1 \). So equation \( y = x + 2 \) (since when \( x = -2 \), \( y = 0 \): \( 0 = -2 + 2 \), correct). Then, after \( x = 6 \), the line is decreasing. Let's find the equation of the decreasing line. Let's take two points: \( (6, 8) \) and let's say \( (8, 6) \) (since the arrow is going down). So slope is \( \frac{6 - 8}{8 - 6} = -1 \). So equation is \( y - 8 = -1(x - 6) \), so \( y = -x + 14 \). Now, for \( x = 7 \), plug into this equation: \( y = -7 + 14 = 7 \). Wait, but maybe simpler: from \( x = 6 \) ( \( y = 8 \)) to \( x = 7 \), since the slope is -1, so \( y \) decreases by 1. So \( 8 - 1 = 7 \). So \( f(7) = 7 \).

Answer:

\( 7 \)