Question

the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = 1$.

Explanation:

Step1: Understand the problem

We need to find the \( x \)-values where \( f(x) = 1 \) from the graph of \( y = f(x) \). This means we look for the points on the graph where the \( y \)-coordinate is 1 and then find the corresponding \( x \)-coordinates.

Step2: Analyze the graph

Looking at the graph, we can see that the parabola (the graph of \( f(x) \)) is symmetric about the vertical line \( x = -3 \) (since the vertex is at \( x = -3 \)). We need to find the \( x \)-values when \( y = 1 \).

From the graph, we can observe that the points where \( y = 1 \) are symmetric with respect to the line \( x=-3 \). Let's find the horizontal distance from the vertex \( x = -3 \) to the points where \( y = 1 \).

Looking at the grid, we can see that when \( y = 1 \), the \( x \)-values are \( x=-4 \) and \( x = -2 \)? Wait, no, let's check again. Wait, the vertex is at \( x=-3 \), and let's see the \( y \)-value at the vertex. The vertex has a \( y \)-value of, let's see, the grid lines: the vertex is at \( y = 1 \)? Wait, no, the grid: each square is 1 unit. The vertex is at \( y = 1 \)? Wait, the graph: the minimum point (vertex) is at \( y = 1 \)? Wait, no, looking at the graph, the vertex is at \( y = 1 \)? Wait, the \( y \)-axis: the vertex is at \( x=-3 \), and the \( y \)-coordinate of the vertex is 1? Wait, no, the \( y \)-axis: the vertex is at \( y = 1 \)? Wait, the graph shows that the vertex is at \( ( - 3,1) \)? Wait, no, let's check the grid. The \( y \)-axis: from \( y = 0 \) (x-axis) up, the first grid line is \( y = 1 \), then \( y = 2 \), etc. The vertex is at \( y = 1 \)? Wait, no, the graph: the curve comes down to a minimum at \( y = 1 \), at \( x=-3 \)? Wait, no, the graph: when \( x=-3 \), the \( y \)-value is 1? Wait, no, the graph: the vertex is at \( ( - 3,1) \)? Wait, no, looking at the graph, the vertex is at \( x=-3 \), and the \( y \)-coordinate of the vertex is 1? Wait, no, the \( y \)-axis: the vertex is at \( y = 1 \), so the vertex is \( ( - 3,1) \). Wait, that would mean that the minimum value of \( f(x) \) is 1, so the only point where \( f(x)=1 \) is the vertex? But that can't be, because the graph is a parabola opening upwards, so the minimum value is 1, so \( f(x)=1 \) only at the vertex \( x=-3 \).

Wait, let's re-examine the graph. The vertex is at \( x=-3 \), and the \( y \)-coordinate of the vertex is 1. So the function \( f(x) \) has a minimum value of 1 at \( x=-3 \). Therefore, the only solution to \( f(x)=1 \) is \( x=-3 \). Wait, that makes sense because if the vertex is the minimum point with \( y = 1 \), then the only point where \( f(x)=1 \) is the vertex, since the parabola opens upwards (so it never goes below \( y = 1 \)).

Wait, let's check the graph again. The graph starts from the top left, comes down to a minimum at \( ( - 3,1) \), then goes up. So the minimum value of \( f(x) \) is 1, achieved only at \( x=-3 \). Therefore, the only \( x \)-value where \( f(x)=1 \) is \( x=-3 \).

Answer:

\( x = -3 \)