Question

Question #010fe

Answer 1

Transform the equation to factored form, and answer the following questions.

Explanation:

Basically, what they're asking is to transform the equation to vertex form, and answer the following questions using the now-transformed equation.

`f(x)=1/2x^2+1/2x-10`

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Intercept Form

Intercept form is another way to say factored form. This involves factoring.

First, let's factor out the GCF - `1/2`.

`f(x)=1/2x^2+1/2x-10`

`f(x)=1/2(x^2+x-20)`

Now it's a simple trinomial. "What two numbers added equal `b`, and multiplied equal `c`?"

In this case, it is `5` and `-4`.

`f(x)=1/2(x+5)(x-4)`

This is the factored form (AKA intercept form).

If we expand the brackets, we will get the standard form.

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Zeros

In order to find the `x`-intercepts, the most simplest method is to use the quadratic formula.

`x = (-b+-sqrt(b^2-4ac))/(2a)`

Here, we sub in the values from a standard equation.

`x = (-(1/2)+-sqrt((1/2)^2-4(1/2)(-10)))/(2(1/2))`

We simplify.

`x = (-1/2+-sqrt(81/4))/(1)`

And solve for `x`.

`x [+] = 4`

`x [-] = -5`

If we graph it, we can conclude that the zeros are `4` and `-5`.

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Vertex

To find the vertex, in my opinion, the easiest way is to complete the square.

First, we factor out the `a`-value from the first two terms.

`f(x)=1/2x^2+1/2x-10`

`f(x)=1/2(x^2+x)-10`

Now, we divide the `b`-value by two, and the square it. In the end, we still get one. We must add and subtract those values in order to keep the equation true to itself.

`f(x)=1/2(x^2+x + 1/4 - 1/4)-10`

Now we expand the `-1/4`.

`f(x)=1/2(x^2+x + 1/4) -1/8 -10`

Now we simplify the equation.

`f(x)=1/2(x + 1/2)^2 -81/8`

That's our vertex form. If we expand it, we would get the standard form.

Since we have our vertex formula, we can just use the `h` and the `k`-values where `(h,k)` = `(x, y)`.

Thus, the vertex is `(-1/2, -81/2)`.

If we graph it, we can conclude the that vertex is indeed `(-1/2, -81/2)`.

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Axis of Symmetry

The axis of symmetry is basically the `x`-value of the vertex.

Thus, it is: `x=-1/2`.

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Here's a graph.

graph{1/2x^2+1/2x-10 [-10, 10, -5, 5]}

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Hope this helps :)