Questions: Describe the transformations that produce the graph of (g) from the graph of (f). (f(x)=x^2 ; g(x)=-(x-9)^2+8)

Transcript text: Describe the transformations that produce the graph of $g$ from the graph of $f$. \[ f(x)=x^{2} ; g(x)=-(x-9)^{2}+8 \]

Solution

Solution Steps

To transform the graph of $ f(x) = x^2 $ into the graph of $ g(x) = -(x-9)^2 + 8 $, we need to consider the following transformations:

Horizontal shift: The term $ (x-9) $ indicates a shift to the right by 9 units.
Reflection: The negative sign in front of the squared term indicates a reflection across the x-axis.
Vertical shift: The $ +8 $ at the end indicates a shift upwards by 8 units.

Step 1: Identify the Base Function

The base function given is $ f(x) = x^2 $, which is a standard parabola opening upwards with its vertex at the origin $(0, 0)$.

Step 2: Analyze the Transformation Function

The function $ g(x) = -(x-9)^2 + 8 $ can be analyzed by comparing it to the standard form of a transformed quadratic function, which is $ a(x-h)^2 + k $.

Step 3: Determine the Transformations

Horizontal Shift: The term $(x-9)$ indicates a horizontal shift. The graph of $ f(x) = x^2 $ is shifted 9 units to the right. This is because the transformation is in the form $(x-h)$, where $ h = 9 $.
Vertical Shift: The constant term $+8$ indicates a vertical shift. The graph is shifted 8 units upwards.
Reflection: The negative sign in front of the squared term, $-$, indicates a reflection across the x-axis. This changes the direction of the parabola from opening upwards to opening downwards.