Question

homework assignment 3.2: quadratic functions
score: 0/10 answered: 1/10
question 2
write an equation for the quadratic graphed below:
x-intercepts: (-2, 0) and (3, 0); y-intercept: (0, -2)
y =
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Explanation:

Step1: Use factored form of quadratic

The factored form of a quadratic with x - intercepts \(x = r_1\) and \(x = r_2\) is \(y=a(x - r_1)(x - r_2)\). Given x - intercepts \((-2,0)\) and \((3,0)\), so \(r_1=-2\) and \(r_2 = 3\). Then the equation is \(y=a(x + 2)(x - 3)\).

Step2: Substitute y - intercept to find a

The y - intercept is \((0,-2)\), so substitute \(x = 0\) and \(y=-2\) into the equation:
\(-2=a(0 + 2)(0 - 3)\)
\(-2=a(2)(-3)\)
\(-2=-6a\)

Step3: Solve for a

Divide both sides by \(-6\): \(a=\frac{-2}{-6}=\frac{1}{3}\)

Step4: Write the final equation

Substitute \(a=\frac{1}{3}\) back into the factored form: \(y=\frac{1}{3}(x + 2)(x - 3)\). We can also expand it:
First, multiply \((x + 2)(x - 3)=x^{2}-3x + 2x-6=x^{2}-x - 6\). Then \(y=\frac{1}{3}x^{2}-\frac{1}{3}x - 2\)

Answer:

\(y=\frac{1}{3}(x + 2)(x - 3)\) (or \(y=\frac{1}{3}x^{2}-\frac{1}{3}x - 2\))