Determine the equation of ℎ if ℎ(𝑥) = 𝑓(𝑥 − 3). Hence, write down the coordinates of the new turning point of ℎ(𝑥).

To determine the equation of \( h(x) \) given that \( h(x) = f(x - 3) \) where \( f(x) = 2x^2 \), we'll follow these steps:

### Step 1: Substitute \( x - 3 \) into \( f(x) \)
Given \( f(x) = 2x^2 \), we substitute \( x \) with \( x - 3 \) to find \( h(x) \):

\[
h(x) = f(x - 3) = 2(x - 3)^2
\]

### Step 2: Expand the expression
Now, expand \( (x - 3)^2 \):

\[
(x - 3)^2 = x^2 - 6x + 9
\]

Substitute this back into the equation for \( h(x) \):

\[
h(x) = 2(x^2 - 6x + 9)
\]

Finally, distribute the 2:

\[
h(x) = 2x^2 - 12x + 18
\]

So, the equation of \( h(x) \) is:

\[
h(x) = 2x^2 - 12x + 18
\]

### Step 3: Determine the new turning point of \( h(x) \)
The turning point (vertex) of a quadratic function \( ax^2 + bx + c \) can be found using the formula:

\[
x_{\text{vertex}} = \frac{-b}{2a}
\]

For \( h(x) = 2x^2 - 12x + 18 \):
- \( a = 2 \)
- \( b = -12 \)

\[
x_{\text{vertex}} = \frac{-(-12)}{2(2)} = \frac{12}{4} = 3
\]

Now, substitute \( x = 3 \) into \( h(x) \) to find the \( y \)-coordinate of the vertex:

\[
y_{\text{vertex}} = h(3) = 2(3)^2 - 12(3) + 18
\]
\[
y_{\text{vertex}} = 2(9) - 36 + 18 = 18 - 36 + 18 = 0
\]

Thus, the coordinates of the new turning point of \( h(x) \) are \( (3, 0) \).

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### Any further questions or need more details? Feel free to ask!

#### Here are 5 related questions to consider:
1. How does shifting a function horizontally affect its graph?
2. What is the effect of shifting \( f(x) \) vertically instead of horizontally?
3. How would the vertex change if the function were \( h(x) = f(x + 3) \) instead?
4. Can you derive the general formula for the vertex of a quadratic function \( ax^2 + bx + c \)?
5. How do you determine the direction in which a parabola opens based on the coefficient \( a \) in \( ax^2 \)?

#### Tip:
When transforming functions, always consider the impact on key points like the vertex or intercepts. Horizontal shifts \( h(x) = f(x - a) \) move the graph to the right if \( a \) is positive, and to the left if \( a \) is negative.

Explore how to determine the equation of h(x) if h(x) = f(x - 3) with a detailed solution. Learn about quadratic functions and function transformations, specifically horizontal shifts. Suitable for students in Grades 10-12.

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