Question

write an equation for the quadratic graphed below
\\(x\\)-intercepts: \\((-1,0)\\) and \\((4,0)\\). \\(y\\)-intercept: \\((0,2)\\)

\\(y = \\)

Explanation:

🆕 New Concept Discovered: Writing Quadratic Equations from Intercepts
Using intercepts to build a quadratic function

Step 1: Use intercept form

The intercept form of a quadratic equation is:

$$ y = a(x - p)(x - q) $$

where \( p \) and \( q \) are the \( x \)-intercepts. Given the \( x \)-intercepts are \( (-1,0) \) and \( (4,0) \):

$$ p = -1, \quad q = 4 $$

Substitute these values into the formula:

$$ y = a(x - (-1))(x - 4) $$

$$ y = a(x + 1)(x - 4) $$

Step 2: Solve for the leading coefficient

Use the given \( y \)-intercept \( (0,2) \) to find the value of \( a \). Substitute \( x = 0 \) and \( y = 2 \) into the equation:

$$ 2 = a(0 + 1)(0 - 4) $$

$$ 2 = a(1)(-4) $$

$$ 2 = -4a $$

$$ a = \frac{2}{-4} = -\frac{1}{2} $$

Step 3: Write the final equation

Substitute \( a = -\frac{1}{2} \) back into the intercept form equation:

$$ y = -\frac{1}{2}(x + 1)(x - 4) $$

Answer:

$$ y = -\frac{1}{2}(x + 1)(x - 4) $$