QuestionWrite an equation (any form) for the quadratic graphed below $y=\square$

Studdy Solution

STEP 1

1. The quadratic function has a vertex form because the vertex is given.
2. The vertex form of a quadratic equation is $y = a(x-h)^2 + k$ , where $(h, k)$ is the vertex.
3. The vertex of the parabola is at $(-3, -4)$ .
4. The parabola intersects the y-axis at $(0, 5)$ , which provides another point on the graph to solve for $a$ .

STEP 2

1. Write the equation in vertex form using the vertex.
2. Substitute the y-intercept point into the equation to solve for $a$ .
3. Write the final equation with the calculated value of $a$ .

STEP 3

Write the equation in vertex form using the vertex $(-3, -4)$ . The vertex form is:
$y = a(x - h)^2 + k$
Substitute $h = -3$ and $k = -4$ :
$y = a(x + 3)^2 - 4$

STEP 4

Use the y-intercept point $(0, 5)$ to find $a$ . Substitute $x = 0$ and $y = 5$ into the equation:
$5 = a(0 + 3)^2 - 4$

STEP 5

Solve for $a$ :
$5 = a(3)^2 - 4$ $5 = 9a - 4$
Add 4 to both sides:
$9 = 9a$
Divide by 9:
$a = 1$

STEP 6

Write the final equation using the calculated value of $a = 1$ :
$y = 1(x + 3)^2 - 4$
Simplify:
$y = (x + 3)^2 - 4$
The equation of the quadratic function is:
$y = (x + 3)^2 - 4$

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