For the given​ equation, list the intercepts and test for symmetry.

y equals x squared minus 6 x minus 27y=x2−6x−27

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Part 1

What​ is/are the​ intercept(s)? Select the correct choice​ and, if​ necessary, fill in the answer box within your choice.

A.

The​ intercept(s) is/are

enter your response here.

​(Type an ordered pair. Use a comma to separate answers as​ needed.)

To find the intercepts of the given equation \( y = x^2 - 6x - 27 \), we'll determine the points where the graph of the equation crosses the x-axis and the y-axis.

### 1. Finding the x-intercepts
The x-intercepts occur where \( y = 0 \). To find these, set the equation equal to 0 and solve for \( x \):

\[
x^2 - 6x - 27 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( x^2 - 6x - 27 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = -27 \)

Plugging these values into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-27)}}{2 \cdot 1}
\]

\[
x = \frac{6 \pm \sqrt{36 + 108}}{2}
\]

\[
x = \frac{6 \pm \sqrt{144}}{2}
\]

\[
x = \frac{6 \pm 12}{2}
\]

This gives two solutions:

\[
x_1 = \frac{6 + 12}{2} = \frac{18}{2} = 9
\]

\[
x_2 = \frac{6 - 12}{2} = \frac{-6}{2} = -3
\]

So, the x-intercepts are:

\[
(9, 0) \quad \text{and} \quad (-3, 0)
\]

### 2. Finding the y-intercept
The y-intercept occurs where \( x = 0 \). To find this, substitute \( x = 0 \) into the equation:

\[
y = 0^2 - 6 \cdot 0 - 27
\]

\[
y = -27
\]

So, the y-intercept is:

\[
(0, -27)
\]

### 3. Intercepts Summary
The intercepts are:
- **x-intercepts:** \( (9, 0) \) and \( (-3, 0) \)
- **y-intercept:** \( (0, -27) \)

### Part 1 Answer
The intercepts are:  
**(9, 0), (-3, 0), (0, -27)**

Would you like a detailed explanation of how to test for symmetry for this equation, or do you have any other questions?

Here are 5 related questions to further explore this topic:

1. How do you factorize a quadratic equation to find the x-intercepts?
2. What does it mean for a graph to have symmetry with respect to the y-axis?
3. How can you determine the vertex of the parabola given by the equation \(y = x^2 - 6x - 27\)?
4. What is the axis of symmetry for the parabola represented by \(y = x^2 - 6x - 27\)?
5. How do you determine if a quadratic equation will open upwards or downwards?

**Tip:** When solving for intercepts, always remember to substitute the appropriate values (0 for x when finding y-intercepts and 0 for y when finding x-intercepts).

For the given equation, list the intercepts and test for symmetry.

y = x^2 - 6x - 27

Explore how to find the x- and y-intercepts of the quadratic equation y = x^2 - 6x - 27 and test for symmetry. This solution includes detailed steps using the quadratic formula and discusses key algebraic concepts. Suitable for students in Grades 9-10.

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