Question

graph the function $f(x) = 5(x + 4)^2$. plot the vertex. then plot another point on the parabola. if you make a mistake, you can erase your parabola by selecting the second point and placing it on top of the first.

Explanation:

Step1: Identify the vertex form

The function is given in vertex form \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( f(x) = 5(x + 4)^2 \), we can rewrite it as \( f(x) = 5(x - (-4))^2 + 0 \). So the vertex \((h, k)\) is \((-4, 0)\).

Step2: Find another point

Let's choose \( x = -3 \) (a value close to the vertex for easy calculation). Substitute \( x = -3 \) into the function:
\( f(-3) = 5(-3 + 4)^2 = 5(1)^2 = 5 \). So the point \((-3, 5)\) is on the parabola.

Step3: Graphing steps

1. Plot the vertex \((-4, 0)\) on the coordinate plane.
2. Plot the point \((-3, 5)\) (or any other point found using the function).
3. Since the coefficient \( a = 5 \) is positive, the parabola opens upwards. Draw a parabola passing through the vertex and the other point, symmetric about the vertical line \( x = -4 \).

Answer:

The vertex is \((-4, 0)\), and another point (e.g., \((-3, 5)\)) can be used to graph the parabola \( f(x) = 5(x + 4)^2 \), which opens upwards with vertex at \((-4, 0)\).