Question

1. the graph at the right shows $f(x) = 2(x - 3)^2 - 1$.

a) evaluate $f(-2)$.
b) what does $f(-2)$ represent on the graph of $f$?

Explanation:

Part (a)

Step 1: Substitute \( x = -2 \) into the function

We have the function \( f(x) = 2(x - 3)^2 - 1 \). To find \( f(-2) \), we substitute \( x = -2 \) into the function. So we get \( f(-2)=2(-2 - 3)^2 - 1 \).

Step 2: Simplify the expression inside the parentheses

First, simplify \( -2 - 3 \). \( -2-3=-5 \), so the expression becomes \( f(-2)=2(-5)^2 - 1 \).

Step 3: Simplify the exponent

Calculate \( (-5)^2 \). We know that \( (-5)^2=(-5)\times(-5) = 25 \), so now the expression is \( f(-2)=2\times25 - 1 \).

Step 4: Perform the multiplication and subtraction

First, multiply \( 2\times25 = 50 \), then subtract 1: \( 50 - 1=49 \).

On the graph of a function \( y = f(x) \), the value \( f(a) \) represents the \( y \)-coordinate of the point on the graph where the \( x \)-coordinate is \( a \). In the case of \( f(-2) \), \( x=-2 \) and \( f(-2) = 49 \), so \( f(-2) \) represents the \( y \)-value (or the ordinate) of the point on the graph of \( f(x) \) when \( x=-2 \), that is, it represents the \( y \)-coordinate of the point with \( x \)-coordinate \( -2 \) on the graph of \( f \).

Answer:

For part (a), \( f(-2)=49 \)

Part (b)