Question

4 graph the quadratic function.

\\f(x) = \frac{1}{9}(x - 2)(x + 4)\\

Explanation:

⚡ Using what you learned: graphing quadratic functions

Step 1: Find the x-intercepts

The quadratic function is given in factored form:

$$ f(x) = \frac{1}{9}(x - 2)(x + 4) $$

Set \( f(x) = 0 \) to find the \( x \)-intercepts:

$$ x - 2 = 0 \implies x = 2 $$

$$ x + 4 = 0 \implies x = -4 $$

The \( x \)-intercepts are at the points \( (2, 0) \) and \( (-4, 0) \).

Step 2: Find the vertex

The \( x \)-coordinate of the vertex is halfway between the two \( x \)-intercepts:

$$ x_v = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1 $$

Substitute \( x = -1 \) back into the function to find the \( y \)-coordinate of the vertex:

$$ f(-1) = \frac{1}{9}(-1 - 2)(-1 + 4) $$

$$ f(-1) = \frac{1}{9}(-3)(3) $$

$$ f(-1) = \frac{1}{9}(-9) = -1 $$

The vertex is at the point \( (-1, -1) \).

Step 3: Identify key points to plot

To graph the parabola, plot the following key points on the coordinate plane:

• Vertex: \( (-1, -1) \)
• x-intercepts: \( (2, 0) \) and \( (-4, 0) \)

Answer:

To graph the quadratic function:

1. Select the PARABOLA tool.
2. Plot the vertex at \( (-1, -1) \).
3. Plot another point on the curve, such as one of the \( x \)-intercepts at \( (2, 0) \) or \( (-4, 0) \), to define the shape of the parabola opening upwards.