Problem 84 (a)write the equation in standar... [FREE SOLUTION]

Chapter 11: Problem 84

(a)write the equation in standard form and (b) use properties of the standard form to graph the equation. $$ x=-3 y^{2}-6 y-5 $$

Short Answer

Expert verified

The standard form is $ x = -3(y + 1)^2 - 2 $. Graph the vertex at (-2, -1) with the parabola opening to the left.

Step by step solution

- Rewrite the equation

Rewrite the given equation in a form that can be rearranged to the standard form. The equation is: $ x = -3y^2 - 6y - 5 $

- Complete the square

Complete the square for the quadratic terms in $ y $ on the right-hand side. Start by factoring out the coefficient of $ y^2 $: $ x = -3(y^2 + 2y) - 5 $. To complete the square, add and subtract the square of half the coefficient of $ y $ inside the parentheses: $ y^2 + 2y $ becomes $ y^2 + 2y + 1 - 1 $. So, we get: $ x = -3(y^2 + 2y + 1 - 1) - 5 $

- Simplify and rearrange

Simplify the equation by combining like terms and rearranging: $ x = -3((y+1)^2 - 1) - 5 $ Distribute the -3: $ x = -3(y+1)^2 + 3 - 5 $ Finally, combine constants: $ x = -3(y+1)^2 - 2 $

- Identify the standard form

The standard form of a parabolic equation opening horizontally is: $ x = a(y - k)^2 + h $ From the previous step, we can see: $ x = -3(y + 1)^2 - 2 $ Therefore, the equation is already in standard form with $ a = -3, k = -1, h = -2 $

- Graph the equation

Use the properties of the standard form to graph the equation: * The vertex of the parabola is (h, k), which is (-2, -1). * The graph opens to the left (since $ a < 0 $). Plot the vertex at (-2, -1) and draw a parabola opening to the left. The value of $ a $ determines the width of the parabola. Since $ a = -3 $, the parabola will be relatively narrow.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form Equation

A standard form equation for a parabola is a convenient way to express its properties clearly. For a parabola opening horizontally, the standard form is $ x = a(y - k)^2 + h $. The $ a $ value affects the width and direction of the parabola, while $ h $ and $ k $ locate the vertex.In the provided exercise, the equation $x = -3y^2 -6y -5$ needs to be rewritten in standard form. After completing the steps, we get $x = -3(y+1)^2 -2$, confirming the equation in standard form. By understanding this form, it is easier to extract the properties of the parabola.

Parabola

A parabola is a U-shaped curve that can open either up, down, left, or right. The standard form makes it easier to determine the direction:

For a vertical parabola: $ y = a(x-h)^2 + k $
For a horizontal parabola: $ x = a(y - k)^2 + h $

The sign of $ a $ tells us the direction.

If $ a > 0 $, the parabola opens upwards or to the right.
If $ a < 0 $, it opens downwards or to the left.

In our exercise example, since $ a = -3 $, the parabola opens to the left.

Completing the Square

Completing the square is a method used to easily convert a quadratic equation to standard form.Here鈥檚 a step-by-step for our example:

Start with rewriting: $x = -3y^2 -6y -5$.
Factor out the coefficient of $ y^2 $: $x = -3(y^2 + 2y) - 5$.
Add and subtract $ (\frac{2}{2})^2 $ inside the parentheses: $x = -3(y^2 + 2y + 1 - 1) - 5$.
Simplify inside the parentheses: $x = -3((y+1)^2 -1) - 5$.
Distribute $ -3 $ and combine constants: $x = -3(y+1)^2 + 3 - 5 = -3(y+1)^2 -2$.

Graphing Equations

Graphing a parabola becomes straightforward once the equation is in standard form. Here鈥檚 how to graph $ x = -3(y+1)^2 -2 $:

Identify the vertex: From $ h $ and $ k $ in standard form, the vertex is $(-2, -1)$.
Plot the vertex on the graph.
Determine the direction: Since $ a < 0 $, the parabola opens left.
Draw the parabola: Make sure it narrows as $ |a| $ becomes larger.

Following these steps will help visualize the parabolic graph accurately.

Vertex Form

The vertex form of a parabolic equation is specific and useful for identifying the vertex quickly.The general vertex form is $ x = a(y - k)^2 + h $, which clearly shows the vertex at $(h, k)$.For our exercise example, $x = -3(y+1)^2 -2$, gives us the vertex $(-2, -1)$. With this form, you can immediately see the highest or lowest point on the parabola, allowing easier sketching and graphing of the curve.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

(a)write the equation in standard form and (b) use properties of the standard form to graph the equation. $$ x=-3 y^{2}-6 y-5 $$

Short Answer

Step by step solution

- Rewrite the equation

- Complete the square

- Simplify and rearrange

- Identify the standard form

- Graph the equation

Key Concepts

Standard Form Equation

Parabola

Completing the Square

Graphing Equations

Vertex Form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Discrete Mathematics

Statistics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.