Question

graph the function $f(x) = 7(x + 3)^2 - 8$. 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 \( f(x) = 7(x + 3)^2 - 8 \) is in vertex form \( f(x)=a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h=-3 \) (since \( x+3=x - (-3) \)) and \( k = - 8 \), so the vertex is \((-3,-8)\).

Step2: Find another point

Let's choose \( x=-2 \) (one unit to the right of the vertex's x - coordinate). Substitute \( x=-2 \) into the function: \( f(-2)=7(-2 + 3)^2-8=7(1)^2-8=7 - 8=-1 \). So the point \((-2,-1)\) is on the parabola.

Step3: Graphing

Plot the vertex \((-3,-8)\) on the coordinate plane. Then plot the point \((-2,-1)\). Since \( a = 7>0 \), the parabola opens upwards. Draw a parabola passing through these points (and other points if needed, but for the task, plotting the vertex and one more point is sufficient).

Answer:

Vertex: \((-3, - 8)\), Another point: \((-2, - 1)\) (and the parabola is graphed with these points, opening upwards)