Question

due wednesday by 11:59pm points 100 submitting an external tool

Explanation:

To determine the equation of the parabola (assuming the question is to find its equation), we analyze the graph:

Step 1: Identify the vertex and roots

The vertex is at \((-4, 2)\) (the highest point). The roots (x-intercepts) are at \(x = -5\) and \(x = -3\) (since the parabola crosses the x-axis at these points).

Step 2: Write the factored form

A parabola with roots \(r_1\) and \(r_2\) has the form \(y = a(x - r_1)(x - r_2)\). Substituting \(r_1 = -5\) and \(r_2 = -3\):
\[ y = a(x + 5)(x + 3) \]

Step 3: Solve for \(a\) using the vertex

Substitute the vertex \((-4, 2)\) into the equation:
\[ 2 = a(-4 + 5)(-4 + 3) \]
\[ 2 = a(1)(-1) \]
\[ 2 = -a \]
\[ a = -2 \]

Step 4: Expand the equation

Substitute \(a = -2\) into the factored form:
\[ y = -2(x + 5)(x + 3) \]
Expand \((x + 5)(x + 3) = x^2 + 8x + 15\), so:
\[ y = -2(x^2 + 8x + 15) \]
\[ y = -2x^2 - 16x - 30 \]

If the question was to identify the vertex, roots, or another property, adjust accordingly. For example, the vertex is \(\boldsymbol{(-4, 2)}\), and the roots are \(\boldsymbol{x = -5}\) and \(\boldsymbol{x = -3}\).

(Note: The specific answer depends on the exact question, e.g., equation, vertex, roots, etc.)

Answer:

To determine the equation of the parabola (assuming the question is to find its equation), we analyze the graph:

Step 1: Identify the vertex and roots

The vertex is at \((-4, 2)\) (the highest point). The roots (x-intercepts) are at \(x = -5\) and \(x = -3\) (since the parabola crosses the x-axis at these points).

Step 2: Write the factored form

A parabola with roots \(r_1\) and \(r_2\) has the form \(y = a(x - r_1)(x - r_2)\). Substituting \(r_1 = -5\) and \(r_2 = -3\):
\[ y = a(x + 5)(x + 3) \]

Step 3: Solve for \(a\) using the vertex

Substitute the vertex \((-4, 2)\) into the equation:
\[ 2 = a(-4 + 5)(-4 + 3) \]
\[ 2 = a(1)(-1) \]
\[ 2 = -a \]
\[ a = -2 \]

Step 4: Expand the equation

Substitute \(a = -2\) into the factored form:
\[ y = -2(x + 5)(x + 3) \]
Expand \((x + 5)(x + 3) = x^2 + 8x + 15\), so:
\[ y = -2(x^2 + 8x + 15) \]
\[ y = -2x^2 - 16x - 30 \]

If the question was to identify the vertex, roots, or another property, adjust accordingly. For example, the vertex is \(\boldsymbol{(-4, 2)}\), and the roots are \(\boldsymbol{x = -5}\) and \(\boldsymbol{x = -3}\).

(Note: The specific answer depends on the exact question, e.g., equation, vertex, roots, etc.)