Question

sketch the parabola using the given information. vertex (0, 5), point (-7, -9)
use the graphing tool to graph the parabola.

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Given vertex \((0,5)\), so \( h = 0 \), \( k = 5 \). The equation becomes \( y = a(x - 0)^2 + 5 = ax^2 + 5 \).

Step2: Find the value of \( a \)

We know the parabola passes through \((-7, -8)\). Substitute \( x = -7 \), \( y = -8 \) into the equation:
\( -8 = a(-7)^2 + 5 \)
\( -8 = 49a + 5 \)
Subtract 5 from both sides: \( -13 = 49a \)
Solve for \( a \): \( a = -\frac{13}{49} \)
So the equation is \( y = -\frac{13}{49}x^2 + 5 \).

Step3: Sketch the parabola

• The vertex is at \((0,5)\) (on the y - axis).
• Since \( a = -\frac{13}{49}<0 \), the parabola opens downward.
• To sketch, plot the vertex \((0,5)\). Then use the point \((-7, -8)\) (and its mirror image \((7, -8)\) due to symmetry about the y - axis) to draw the parabola. The parabola will curve downward from the vertex, passing through \((-7, -8)\) and \((7, -8)\).

Answer:

The parabola has the equation \( y = -\frac{13}{49}x^2 + 5 \), with vertex at \((0,5)\), opening downward, and passing through \((-7, -8)\) (and \((7, -8)\) by symmetry). To sketch, plot the vertex and the point, then draw the downward - opening parabola symmetric about the y - axis.