Question

Find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. $$ y=-x^{2}+4 $$

Short answer

Intercepts are (0, 4), (2, 0), and (-2, 0).

Step-by-step solution

Find the y-Intercept

To find the y-intercept, set x = 0 and solve for y. Substituting x = 0 into the equation gives: \[ y = -(0)^2 + 4 = 4. \] So, the y-intercept is (0, 4).

Find the x-Intercepts

To find the x-intercepts, set y = 0 and solve for x. Setting the equation to 0, we get: \[ 0 = -x^2 + 4. \] Solving for x: \[ x^2 = 4 \ \rightarrow \ x = \pm 2. \] So, the x-intercepts are (2, 0) and (-2, 0).

Plot the Intercepts

Plot the intercepts on a coordinate plane: (0, 4), (2, 0), and (-2, 0).

Plot Additional Points

To have a more accurate graph, plot additional points. For example, choose x = 1 and x = -1 and find the corresponding y-values:\[ y = -(1)^2 + 4 = 3 \ \rightarrow (1, 3) \]\[ y = -(-1)^2 + 4 = 3 \ \rightarrow (-1, 3) \]Plot these points on the coordinate plane.

Draw the Curve

Draw a smooth curve through the plotted points to complete the graph of the equation.

Label the Intercepts

Label the intercepts on the graph: (0, 4), (2, 0), and (-2, 0).

Key concepts

Intercepts

Intercepts are the points where the graph of an equation crosses the x-axis and y-axis. To find the **y-intercept**, set x to 0 in the equation. For the equation \( y = -x^{2} + 4 \), substituting x = 0 gives: \[ y = -(0)^2 + 4 = 4 \]. Thus, the y-intercept is (0, 4).
To find the **x-intercepts**, set y to 0 and solve the equation for x. For \( y = -x^{2} + 4 \), setting y to 0 gives: \[ 0 = -x^2 + 4 \rightarrow x^2 = 4 \rightarrow x = \pm 2 \]. Hence, the x-intercepts are (2, 0) and (-2, 0).
Labeling the intercepts on your graph makes it easier to understand where the graph touches the axes.

Plotting Points

Plotting points is a crucial step in graphing any function. Start by plotting the intercepts you found earlier: (0, 4), (2, 0), and (-2, 0).
For a more complete graph, you need additional points. Choose values for x and find the corresponding y-values using the equation \( y = -x^{2} + 4 \).

• For x = 1: \[ y = -(1)^2 + 4 = 3 \rightarrow (1, 3) \]
• For x = -1: \[ y = -(-1)^2 + 4 = 3 \rightarrow (-1, 3) \]

These additional points help you see the shape of the graph better.
Plot all these points on the coordinate plane and prepare to draw the curve.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards. The equation \( y = -x^{2} + 4 \) represents a downward-opening parabola because the coefficient of \ x^{2}\ is negative.
When graphing the parabola, start from the y-intercept and make sure the curve passes smoothly through all plotted points. The intercepts will be on opposite sides of the vertex, the highest point in this case.
For the given equation, the vertex is at the y-intercept (0, 4).
Draw a smooth curve through the points (0, 4), (2, 0), (-2, 0), (1, 3), and (-1, 3) to complete your graph. This ensures precision and clarity in portraying the parabola.
Lastly, don't forget to label the intercepts on your graph for better understanding.