Question

homework assignment 3.2: quadratic functions
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question 1
write an equation (any form) for the quadratic graphed below:
y =
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Explanation:

Step1: Identify the vertex and a point

The vertex of the parabola is at \((3, -2)\) (from the graph, the lowest point). Let's use the vertex form of a quadratic function \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. So \(h = 3\), \(k = -2\), so the equation becomes \(y = a(x - 3)^2 - 2\). Now we need to find \(a\). We can use another point on the parabola. Let's take the y - intercept, which is \((0, 6)\) (from the graph, when \(x = 0\), \(y = 6\)).

Step2: Substitute the point into the equation to find \(a\)

Substitute \(x = 0\) and \(y = 6\) into \(y = a(x - 3)^2 - 2\):
\[

$$\begin{align*} 6&=a(0 - 3)^2 - 2\\ 6&=a(9)-2\\ 6 + 2&= 9a\\ 8&= 9a\\ a&=\frac{8}{9} \end{align*}$$

\]

Step3: Write the equation in vertex form

Now that we have \(a=\frac{8}{9}\), \(h = 3\), and \(k=-2\), the vertex form of the quadratic equation is \(y=\frac{8}{9}(x - 3)^2-2\). We can also expand it to standard form if needed, but vertex form is acceptable.

Answer:

\(y=\frac{8}{9}(x - 3)^2-2\) (or other equivalent forms like standard form \(y=\frac{8}{9}x^2-\frac{16}{3}x + 6\))