Chapter 11: Q.83 (page 1101)

In the following exercises, graph each equation by using properties.
$x = - 2 y^{2} - 12 y - 16$

Short Answer

Expert verified

The graph of the equation is

Step by step solution

Step 1. Given Information

Given equation of the parabola is $x = - 2 y^{2} - 12 y - 16$

(a) The given equation has to be determined in standard form.

(b) The graph of the parabola is to be determined using properties.

Step 2. Equation in standard form

The equation of a parabola in standard form $x = a {(y - k)}^{2} + h$

The given equation is $x = - 2 y^{2} - 12 y - 16$

Writing the equation in standard form is

$x = - 2 y^{2} - 12 y - 16 x = - 2 y^{2} - 12 y - 18 + 2 x = - 2 (y^{2} + 6 y + 9) + 2 x = - 2 {(y + 3)}^{2} + 2$

Here $a = - 2, k = - 3, h = 2$

Since a is negative, the parabola opens to the left.

Step 3. Axis of symmetry

The axis of symmetry of the parabola $x = a {(y - k)}^{2} + h$ is $y = k$

Plugging the values:

$y = - 3$

Hence the axis of symmetry is $y = - 3$

Step 4. Vertex of a parabola

The vertex of the parabola $x = a {(y - k)}^{2} + h$ is $(h, k)$

The vertex of the given parabola is $(2, - 3)$

Step 5. Intercepts

x-intercept is the point where the graph crosses the x-axis and y is 0.

y-intercept is the point where the graph crosses the y-axis and x is 0.

x-intercept:

Plugging $y = 0$ in the equation:

$x = - 2 {(y + 3)}^{2} + 2 x = - 2 {(0 + 3)}^{2} + 2 x = - 2 {(3)}^{2} + 2 x = - 2 {(9)}^{} + 2 x = - 18 + 2 x = - 16$

Hence x-intercept is $(- 16, 0)$

y-intercept:

Plugging $x = 0$ in the equation:

$x = - 2 {(y + 3)}^{2} + 2 0 = - 2 {(y + 3)}^{2} + 2 - 2 {(y + 3)}^{2} = - 2 2 {(y + 3)}^{2} = 2 {(y + 3)}^{2} = 1 y + 3 = \sqrt{1} y + 3 = \pm 1 y = \pm 1 - 3 y = 1 - 3, y = - 1 - 3 y = - 2, y = - 4$

Hence y-intercepts are $(0, - 2), (0, - 4)$

Step 6. Graphing the parabola

Using the properties, plotting the points and graphing the parabola:

Conclusion

The graph of the parabola is

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In the following exercises, graph each equation by using properties.
$x = - 2 y^{2} - 12 y - 16$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Equation in standard form

Step 3. Axis of symmetry

Step 4. Vertex of a parabola

Step 5. Intercepts

Step 6. Graphing the parabola

Conclusion

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In the following exercises, graph each equation by using properties.x=-2y2-12y-16

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Equation in standard form

Step 3. Axis of symmetry

Step 4. Vertex of a parabola

Step 5. Intercepts

Step 6. Graphing the parabola

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In the following exercises, graph each equation by using properties.
$x = - 2 y^{2} - 12 y - 16$