/*! 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 25 Write the quadratic function in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 25

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}+34 x+289$$

Short Answer

Expert verified
The standard form of the quadratic function is \(f(x) = (x + 17)^2\). The vertex of the parabola is (-17, 0), the axis of symmetry is \(x = -17\), and the x-intercept is (-17, 0).

Step by step solution

01

Convert to standard form

We'll start by rewriting \(f(x) = x^2 + 34x + 289\) in standard (vertex) form, which is \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. The given quadratic function is a perfect square trinomial, that is \(f(x) = (x + 17)^2\). This is now in standard form.
02

Identify the vertex and axis of symmetry

We will identify the vertex of the parabola by comparing our equation to the vertex form. The vertex is given by the point \((h, k)\), thus our vertex is \((-17, 0)\). The axis of symmetry is a vertical line that passes through the vertex. Therefore, our axis of symmetry is \(x = -17\).
03

Find the x-intercepts

The x-intercepts are the points where the parabola intersects the x-axis. This happens when the y-value (or \(f(x)\) ) is 0. Hence, to find the x-intercepts, we set \(f(x) = 0\). Hence, \(0 = (x + 17)^2\). Solving for x gives \(x = -17\). So, the x-intercept is \((-17, 0)\). However, please consider that this quadratic function has exactly one x-intercept, as it's a perfect square trinomial, and it touches but does not cross the x-axis.

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

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-4 x^{3}+5 x^{2}-2 x-6\) (Hint: One factor is \(\left.x^{2}-2 x-2 .\right)\)

Use the position equation $$s=-16 t^{2}+v_{0} t+s_{0}$$ where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?

Decide whether the statement is true or false. Justify your answer. If \(x=-i\) is a zero of the function \(f(x)=x^{3}+i x^{2}+i x-1\) then \(x=i\) must also be a zero of \(f\)

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.