Question

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. $$ y=x^{2}-9 $$

Short answer

The y-intercept is (0, -9) and the x-intercepts are (3, 0) and (-3, 0).

Step-by-step solution

- Find the y-intercept

To find the y-intercept, set x = 0 and solve for y. This gives the point where the graph intersects the y-axis.When x = 0:\[ y = 0^2 - 9 = -9 \]Thus, the y-intercept is (0, -9).

- Find the x-intercepts

To find the x-intercepts, set y = 0 and solve for x. This gives the points where the graph intersects the x-axis.When y = 0:\[ 0 = x^2 - 9 \]This can be rewritten as:\[ x^2 = 9 \]Taking the square root of both sides:\[ x = \text{±}3 \]Thus, the x-intercepts are (3, 0) and (-3, 0).

- Plot the intercepts

Plot the y-intercept (0, -9) and the x-intercepts (3, 0) and (-3, 0) on a coordinate plane. Label each intercept accordingly.

- Plot additional points and draw the graph

To accurately draw the graph, choose additional values for x and solve for y. For example:When x = 1:\[ y = 1^2 - 9 = -8 \]When x = -1:\[ y = (-1)^2 - 9 = -8 \]When x = 2:\[ y = 2^2 - 9 = -5 \]When x = -2:\[ y = (-2)^2 - 9 = -5 \]Plot these points on the coordinate plane: (1, -8), (-1, -8), (2, -5), and (-2, -5). Connect the points to form the parabola, ensuring it opens upwards.

Key concepts

y-intercept

The y-intercept is the point where the parabola crosses the y-axis. To find the y-intercept of a quadratic equation like \(y = x^2 - 9\), set \(x = 0\). Replace \(x \) with 0 and solve for \(y\). This will give you the coordinates of the intercept.

In this case, setting \ x = 0\: \ y = 0^2 - 9 = -9\. So, the y-intercept is \(0, -9\).

Here's a step-by-step guide to making sure you get it right:

• Set \(x \) to 0 in your equation.
• Solve for \(y \).
• Write down the coordinates as \ (0, y) \.

When graphing, be sure to label this point clearly on the y-axis!

x-intercept

The x-intercepts are where the parabola intersects the x-axis. To find these points, set \(y = 0\) in the quadratic equation and solve for \(x\). This gives you the points where the graph touches or crosses the x-axis.

In our example \ y = x^2 - 9 \, set \(y = 0\), leading to:
\ 0 = x^2 - 9 \.

• Add 9 to both sides: \ x^2 = 9 \.
• Take the square root of both sides to get: \ x = \pm3 \.

This results in two x-intercepts: \(3, 0\) and \(-3, 0\). Always plot these points on the graph and label them, as they are crucial for accurately sketching the parabola.

parabola

A parabola is the U-shaped graph of a quadratic equation like \(y = x^2 - 9\). Parabolas have distinctive features like a vertex (the highest or lowest point) and symmetry around a vertical axis.

Here’s how to graph the parabola step-by-step:

• Find and plot the y-intercept and x-intercepts as described above.
• Choose additional points to ensure accuracy. For instance, substitute \ x = 1 \ and \ x = -1 \ into the equation.
• Calculate their corresponding \ y \ values.

For example:
When \ x = 1\, \ y = 1^2 - 9 = -8\.
When \ x = -1 \, \ y = (-1)^2 - 9 = -8\.

Plot these points, then connect all points smoothly. The curve should open upwards because the coefficient of \(x^2\) is positive. Adjust the graph more accurately by choosing more points if needed.