Find the general form of the quadratic function with the vertex $$ (2,4) $$ and a point on the graph $$ (-4,40) $$.

Question

UpStudy AI · Accepted Answer

We begin with the vertex form of a quadratic function:

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

Given the vertex $$ (2,4) $$, substitute $$ h = 2 $$ and $$ k = 4 $$:

$$
y = a(x - 2)^2 + 4
$$

Next, we use the point $$ (-4,40) $$ that lies on the graph. Substitute $$ x = -4 $$ and $$ y = 40 $$ into the equation:

$$
40 = a(-4 - 2)^2 + 4
$$

Simplify inside the parentheses:

$$
-4 - 2 = -6
$$

Square $$-6$$:

$$
(-6)^2 = 36
$$

Now the equation becomes:

$$
40 = 36a + 4
$$

Subtract $$ 4 $$ from both sides to isolate the term with $$ a $$:

$$
36 = 36a
$$

Solve for $$ a $$:

$$
a = \frac{36}{36} = 1
$$

Now substitute $$ a = 1 $$ back into the vertex form:

$$
y = (x - 2)^2 + 4
$$

To express the function in the general quadratic form, expand the square:

$$
(x - 2)^2 = x^2 - 4x + 4
$$

Then add $$ 4 $$:

$$
y = x^2 - 4x + 4 + 4 = x^2 - 4x + 8
$$

Thus, the quadratic function in general form is:

$$
y = x^2 - 4x + 8
$$
The quadratic function is $$ y = x^2 - 4x + 8 $$.

Find the general form of the quadratic function with the vertex and a point on the graph .