Question

write an equation (any form) for the quadratic graphed below

\\(y = \\)

Explanation:

⚡ Using what you learned: graphing quadratic functions

Step 1: Identify the vertex

The vertex of the parabola is its lowest point:

$$ (h, k) = (3, 0) $$

Step 2: Identify another point on the graph

The graph passes through the y-intercept:

$$ (x, y) = (0, 4.5) $$

It also passes through:

$$ (1, 2) $$

$$ (5, 2) $$

Step 3: Use the vertex form to find \( a \)

The vertex form of a quadratic equation is:

$$ y = a(x - h)^2 + k $$

Substitute the vertex \( (3, 0) \) into the equation:

$$ y = a(x - 3)^2 + 0 $$

$$ y = a(x - 3)^2 $$

Substitute the point \( (1, 2) \) to solve for \( a \):

$$ 2 = a(1 - 3)^2 $$

$$ 2 = a(-2)^2 $$

$$ 2 = 4a $$

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

Step 4: Write the final equation

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

Answer:

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