Question

question
the graph of $y = f(x)$ is the solid black graph below. which function represents the dotted graph?

answer
$y = f(-x) - 3$ $y = -f(x + 3)$
$y = -f(x - 3)$ $y = f(-x) + 3$
submit answer

Explanation:

Step1: Analyze reflection and shift

The solid graph is a parabola opening upwards with vertex at \((0,0)\). The dotted graph is a transformation: first, check reflection (over y - axis or x - axis) and horizontal/vertical shift. The dotted graph seems to be a reflection over the y - axis? Wait, no, let's check the vertex. The solid vertex is \((0,0)\), dotted vertex: let's see the position. Wait, the dotted graph is a parabola opening downwards? Wait, no, the dotted lines: wait, the solid is \(y = f(x)\) (upward parabola at origin). The dotted graph: let's check the transformation. Let's analyze each option:

Option 1: \(y=f(-x)-3\): reflection over y - axis, then down 3.

Option 2: \(y=-f(x + 3)\): reflection over x - axis, shift left 3.

Option 3: \(y=-f(x - 3)\): reflection over x - axis, shift right 3.

Option 4: \(y=f(-x)+3\): reflection over y - axis, shift up 3.

Wait, the solid graph is \(y = f(x)\) (vertex at (0,0), upward). The dotted graph: let's see the vertex. Wait, the dotted lines: looking at the graph, the dotted graph is a reflection over the y - axis? No, wait, the solid is symmetric about y - axis. Wait, maybe the dotted is a reflection over x - axis? No, the dotted lines: wait, the solid is \(y = f(x)\) (upward), the dotted: let's check the vertex. Wait, the solid vertex is (0,0). The dotted graph: let's see the horizontal shift. Wait, the dotted graph's vertex: let's see the x - coordinate. Wait, the solid is at x = 0. The dotted graph: if we look at the transformation, the dotted graph is a reflection over the y - axis? No, wait, maybe horizontal shift and reflection. Wait, the correct transformation: let's check the vertex. The solid vertex is (0,0). The dotted graph: let's see the position. Wait, the dotted graph is a parabola that is a reflection over the y - axis? No, wait, the solid is \(y = f(x)\) (let's assume \(f(x)=x^{2}\) for example). Then:

- \(y = f(-x)=(-x)^{2}=x^{2}\), same as \(f(x)\) (symmetric over y - axis).

- \(y=-f(x)\) is reflection over x - axis (\(y=-x^{2}\)).

Now, let's check the shift. The dotted graph: let's see the vertex. The solid vertex is (0,0). The dotted graph's vertex: looking at the graph, the dotted graph is shifted? Wait, no, maybe the dotted graph is \(y=-f(x + 3)\)? Wait, no, let's re - examine.

Wait, the solid graph is \(y = f(x)\) (vertex at (0,0)). The dotted graph: let's see the direction. The solid is upward, dotted: if we take \(y=-f(x + 3)\), that would be reflection over x - axis (so downward opening) and shift left 3. Let's check the vertex: for \(y=-f(x + 3)\), the vertex is at \(x+3 = 0\Rightarrow x=-3\), \(y = 0\). So vertex at (-3, 0), and opening downward (since -f). Let's see the graph: the dotted lines have vertex at (-3, 0)? Wait, the solid is at (0,0), the dotted graph's vertex: looking at the graph, the dotted lines seem to have vertex at (-3, 0) and opening downward. So \(y=-f(x + 3)\) would be reflection over x - axis (so opening downward) and shift left 3 (vertex at (-3, 0)). Let's verify:

If \(f(x)=x^{2}\) (solid graph), then \(y=-f(x + 3)=-(x + 3)^{2}\), which is a downward - opening parabola with vertex at (-3, 0), which matches the dotted graph's appearance (vertex at (-3, 0) and opening downward).

Now check other options:

- \(y=f(-x)-3\): \(f(-x)=(-x)^{2}=x^{2}\), so \(y=x^{2}-3\), opening upward, vertex at (0, - 3) → not matching.

- \(y=-f(x - 3)=-(x - 3)^{2}\), vertex at (3, 0), opening downward → vertex at (3,0), not matching.

- \(y=f(-x)+3=x^{2}+3\), opening upward, vertex at (0,3) → not matching.

So the correct option is \(y=-f(x + 3)\).

Step2: Confirm the transformation

The solid graph \(y = f(x)\) (upward, vertex (0,0)) becomes \(y=-f(x + 3)\) which is a reflection over the x - axis (so opening downward) and a horizontal shift left by 3 units (vertex at (-3, 0)), which matches the dotted graph's position and direction.

Answer:

\(y=-f(x + 3)\)