Question

1. how does the graph of $f(x) = -3(x - 5)^2 + 7$ compare to the graph of the parent function?

Explanation:

The parent function of a quadratic function is \( y = x^2 \), which is a parabola opening upwards with vertex at the origin \((0,0)\).

Step 1: Analyze the coefficient of the squared term

For the function \( f(x)=-3(x - 5)^2+7 \), the coefficient of \((x - 5)^2\) is \(- 3\).

• The absolute value of the coefficient (\(\vert-3\vert = 3\)) is greater than \(1\), so the graph is vertically stretched by a factor of \(3\) compared to the parent function \(y=x^2\).
• The negative sign of the coefficient (\(-3\)) means the parabola is reflected over the \(x\)-axis (opens downwards) compared to the parent function which opens upwards.

Step 2: Analyze the horizontal shift

The expression inside the square is \((x - 5)\). For a quadratic function in the form \(y=a(x - h)^2+k\), the horizontal shift is determined by \(h\).

• Here \(h = 5\), so the graph is shifted \(5\) units to the right compared to the parent function \(y = x^2\) (which has \(h = 0\)).

Step 3: Analyze the vertical shift

The constant term added at the end is \(+7\). For a quadratic function in the form \(y=a(x - h)^2+k\), the vertical shift is determined by \(k\).

• Here \(k=7\), so the graph is shifted \(7\) units up compared to the parent function \(y=x^2\) (which has \(k = 0\)).

Answer:

The graph of \(f(x)=-3(x - 5)^2+7\) is a vertical stretch by a factor of \(3\), a reflection over the \(x\)-axis, a horizontal shift \(5\) units to the right, and a vertical shift \(7\) units up of the graph of the parent function \(y = x^2\).