Question

1. use the points or graph to write a quadratics equation in the specified form. (complete 2/3)

a. write the intercept form.
(-4,0), (0,0) and (-2,20)
b. write the vertex form.
vertex: (-6, -100) and (0,44)
c. write the standard form.

Explanation:

Part a: Intercept Form

The intercept form of a quadratic equation is \( y = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots (x-intercepts). From the points \((-4, 0)\) and \((0, 0)\), the roots are \( r_1 = -4 \) and \( r_2 = 0 \). So the equation is \( y = a(x + 4)(x - 0) = a x(x + 4) \). Now we use the point \((-2, 20)\) to find \( a \). Substitute \( x = -2 \) and \( y = 20 \) into the equation:

Step 1: Substitute the point into the equation

\( 20 = a(-2)(-2 + 4) \)

Step 2: Simplify the right side

\( 20 = a(-2)(2) = -4a \)

Step 3: Solve for \( a \)

\( a = \frac{20}{-4} = -5 \)
So the intercept form is \( y = -5x(x + 4) \) or \( y = -5x(x - (-4)) \) (more formally \( y = -5x(x + 4) \)).

Part b: Vertex Form

The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. The vertex is \((-6, -100)\), so \( h = -6 \) and \( k = -100 \). The equation is \( y = a(x + 6)^2 - 100 \). We use the point \((0, 44)\) to find \( a \). Substitute \( x = 0 \) and \( y = 44 \) into the equation:

Step 1: Substitute the point into the equation

\( 44 = a(0 + 6)^2 - 100 \)

Step 2: Simplify the right side

\( 44 = 36a - 100 \)

Step 3: Solve for \( a \)

Add 100 to both sides: \( 44 + 100 = 36a \) → \( 144 = 36a \) → \( a = \frac{144}{36} = 4 \)
So the vertex form is \( y = 4(x + 6)^2 - 100 \).

Part c: Standard Form (from the graph or vertex form)

(using vertex form from part b):
The vertex form is \( y = 4(x + 6)^2 - 100 \). We expand this to standard form \( y = ax^2 + bx + c \).

Step 1: Expand \( (x + 6)^2 \)

\( (x + 6)^2 = x^2 + 12x + 36 \)

Step 2: Multiply by 4

\( 4(x^2 + 12x + 36) = 4x^2 + 48x + 144 \)

Step 3: Subtract 100

\( y = 4x^2 + 48x + 144 - 100 = 4x^2 + 48x + 44 \)

Answer:

s:
a. Intercept form: \( \boldsymbol{y = -5x(x + 4)} \) (or \( y = -5x^2 - 20x \) if expanded, but intercept form is factored)
b. Vertex form: \( \boldsymbol{y = 4(x + 6)^2 - 100} \)
c. Standard form: \( \boldsymbol{y = 4x^2 + 48x + 44} \)