/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 83 Parabola vertex property Prove t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 83

Parabola vertex property Prove that if a parabola crosses the \(x\) -axis twice, the \(x\) -coordinate of the vertex of the parabola is halfway between the \(x\) -intercepts.

Short Answer

Expert verified
Question: Prove that the x-coordinate of the vertex of a parabola is the midpoint of the x-intercepts. Answer: To prove this, we wrote the equation of the parabola in vertex form and found the x-intercepts. We then solved for the values of a and k in terms of x-intercepts and h, showing that the x-coordinate of the vertex (h) is the average of the x-intercepts: h = (x1+x2) / 2. This confirms that the x-coordinate of the vertex of the parabola is halfway between the x-intercepts.

Step by step solution

01

Write the parabola in vertex form

The equation of a parabola in vertex form is given by: $$ y = a(x-h)^2 + k $$ where \((h, k)\) is the vertex of the parabola, and \(a\) is a constant.
02

Find the x-intercepts of the parabola

To find the x-intercepts, set the y-value to 0 and solve for x: $$ 0 = a(x-h)^2 + k $$ Let's call the x-intercepts \(x_1\) and \(x_2\): $$ 0 = a(x_1-h)^2 + k \\ 0 = a(x_2-h)^2 + k $$
03

Solve for a and k in terms of x-intercepts and h

We'll now solve the system of equations for a and k in terms of \(x_1, x_2,\) and \(h\): $$ a(x_1-h)^2 = -k \\ a(x_2-h)^2 = -k $$ Dividing the two equations, we get: $$ \frac{(x_1-h)^2}{(x_2-h)^2} = 1 $$ Taking the square root of both sides: $$ \frac{x_1-h}{x_2-h} = \pm 1 $$
04

Solve for h in terms of x-intercepts

Now solve for \(h\): $$ x_1-h = \pm (x_2-h) $$ Add \(h\) to both sides and divide by 2 to get: $$ h = \frac{x_1 + x_2}{2} $$
05

Conclusion

We found that the x-coordinate of the vertex \(h\) is the average of the x-intercepts \(x_1\) and \(x_2\): $$ h = \frac{x_1+x_2}{2} $$ This proves that the x-coordinate of the vertex of the parabola is halfway between the x-intercepts.

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!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. $$\log _{6} 60$$

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed, distance \(=\)speed time elapsed.A function \(y=f(x)\) such that \(y\) is 1 less than the cube of \(x\).

The floor function, or greatest integer function, \(f(x)=\lfloor x\rfloor,\) gives the greatest integer less than or equal to \(x\) Graph the floor function, for \(-3 \leq x \leq 3\).

Approaching a lighthouse A boat approaches a 50 -ft-high lighthouse whose base is at sea level. Let \(d\) be the distance between the boat and the base of the lighthouse. Let \(L\) be the distance between the boat and the top of the lighthouse. Let \(\theta\) be the angle of elevation between the boat and the top of the lighthouse. a. Express \(d\) as a function of \(\theta\) b. Express \(L\) as a function of \(\theta\)

Use the following steps to prove that \(\log _{b} x y=\log _{b} x+\log _{b} y\) a. Let \(x=b^{p}\) and \(y=b^{q} .\) Solve these expressions for \(p\) and \(q\) respectively. b. Use property E1 for exponents to express \(x y\) in terms of \(b, p\) and \(q\) c. Compute \(\log _{b} x y\) and simplify.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.