Find the intercepts and graph the equation by plotting points.
yequalsnegative x squared plus 1
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Part 1
What are the​ intercepts? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ intercept(s) is/are

enter your response here.
​(Use a comma to separate answers as needed. Type an ordered​ pair.)
B.
There are no intercepts.
Part 2
Use the graphing tool to graph the equation.

To solve this problem, we'll break it down into two parts: finding the intercepts and plotting the graph of the equation.

### Part 1: Finding the Intercepts

The given equation is:
\[
y = -x^2 + 1
\]

**1. ** *Finding the y-intercept*:

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:
\[
y = - (0)^2 + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).

**2. ** *Finding the x-intercepts*:

The x-intercepts occur where \( y = 0 \). Set \( y = 0 \) and solve for \( x \):
\[
0 = -x^2 + 1
\]
\[
x^2 = 1
\]
\[
x = \pm 1
\]
So, the x-intercepts are \( (1, 0) \) and \( (-1, 0) \).

Thus, the intercepts are: \((0, 1)\), \((1, 0)\), and \((-1, 0)\).

### Part 2: Graphing the Equation

To graph the equation \( y = -x^2 + 1 \), we can plot several points and draw the curve. Here's a table of values:

\[
\begin{array}{c|c}
x & y \\
\hline
-2 & -(-2)^2 + 1 = -3 \\
-1 & -(-1)^2 + 1 = 0 \\
0 & -(0)^2 + 1 = 1 \\
1 & -(1)^2 + 1 = 0 \\
2 & -(2)^2 + 1 = -3 \\
\end{array}
\]

The points to plot are: \((-2, -3)\), \((-1, 0)\), \((0, 1)\), \((1, 0)\), and \((2, -3)\).

#### Graphing:
The graph is a parabola that opens downward (because of the negative sign before \( x^2 \)) with the vertex at \( (0, 1) \). The x-intercepts are at \( (-1, 0) \) and \( (1, 0) \), and the y-intercept is at \( (0, 1) \).

Would you like a detailed step-by-step explanation on how to graph this equation or have any other questions?

Here are 5 related questions:
1. How do you find the vertex of a quadratic equation?
2. What is the axis of symmetry for the parabola \( y = -x^2 + 1 \)?
3. How do you determine the direction in which a parabola opens?
4. Can you explain how to find the maximum or minimum value of a quadratic function?
5. How do the coefficients of a quadratic equation affect the shape of its graph?

**Tip:** For any quadratic equation of the form \( y = ax^2 + bx + c \), the graph will be a parabola. The sign of \( a \) determines whether it opens upwards (positive \( a \)) or downwards (negative \( a \)).

Learn how to find the intercepts and graph the quadratic equation -y = x^2 + 1 with detailed step-by-step solution. This problem covers key concepts in quadratic equations, including intercepts and graphing. Suitable for students in Grades 9-12.

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