Question

Math

Posted 4 months ago

```
Write equation in translation form.
$f(4)=6$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Identify the vertex form of a parabola. The vertex form of a parabola is given by $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex

step 2

Given the vertex $(1, -2)$, we substitute $h = 1$ and $k = -2$ into the vertex form: $y = a(x - 1)^2 - 2$

step 3

Use the y-intercept $(0, 0)$ to find $a$. Substitute $x = 0$ and $y = 0$ into the equation: $0 = a(0 - 1)^2 - 2$

step 4

Solve for $a$: $0 = a(1) - 2 \implies a = 2$

step 5

Substitute $a = 2$ back into the vertex form: $y = 2(x - 1)^2 - 2$

step 6

Verify the x-intercepts $(-2, 0)$ and $(3, 0)$ by solving $0 = 2(x - 1)^2 - 2$

step 7

Solve $0 = 2(x - 1)^2 - 2$: $2(x - 1)^2 = 2 \implies (x - 1)^2 = 1 \implies x - 1 = \pm 1 \implies x = 2 \pm 1 \implies x = -2 \text{ or } x = 3$

step 8

Confirm the function value $f(4) = 6$ by substituting $x = 4$ into the equation: $f(4) = 2(4 - 1)^2 - 2 = 2(3)^2 - 2 = 2(9) - 2 = 18 - 2 = 16$

Answer

The equation of the parabola in translation form is $y = 2(x - 1)^2 - 2$.

Key Concept

Vertex form of a parabola

Explanation

The vertex form of a parabola is useful for identifying the vertex and transforming the equation to match given points.

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