22) Given the equation $$ y=2(x-5)^{2}-2 $$, find the following: Standard Form: Intercept Form:

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UpStudy AI · Accepted Answer

**Step 1. Write the given equation in vertex form**

We start with the given equation:
$$
y=2(x-5)^2-2
$$

**Step 2. Convert to Standard Form**

To convert it to standard form, expand $$(x-5)^2$$:
$$
(x-5)^2 = x^2 - 10x + 25
$$
Multiply by 2:
$$
2(x-5)^2 = 2x^2 - 20x + 50
$$
Then subtract 2:
$$
y = 2x^2 - 20x + 50 - 2 = 2x^2 - 20x + 48
$$
Thus, the standard form is:
$$
y = 2x^2 - 20x + 48
$$

**Step 3. Find the Intercept Form**

The intercept form of a quadratic function is generally written as:
$$
y = a(x - r_1)(x - r_2)
$$
where $$r_1$$ and $$r_2$$ are the x-intercepts.

Set $$y = 0$$ in the original equation:
$$
0 = 2(x-5)^2 - 2
$$
Add 2 to both sides:
$$
2(x-5)^2 = 2
$$
Divide both sides by 2:
$$
(x-5)^2 = 1
$$
Take the square root of both sides:
$$
x-5 = \pm 1
$$
Solve for $$x$$:
$$
x = 5 \pm 1
$$
Thus, the x-intercepts are:
$$
x = 6 \quad \text{and} \quad x = 4
$$

Now, substitute $$r_1 = 6$$ and $$r_2 = 4$$ into the intercept form:
$$
y = 2(x-6)(x-4)
$$

**Final Answers:**

- **Standard Form:** 
  $$
  y=2x^2-20x+48
  $$

- **Intercept Form:** 
  $$
  y=2(x-6)(x-4)
  $$
**Standard Form:** $$ y = 2x^2 - 20x + 48 $$ **Intercept Form:** $$ y = 2(x - 6)(x - 4) $$

Given the equation , find
the following:
Standard Form:
Intercept Form:

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