/*! 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 21 Finding Intercepts In Exercises ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 21

Finding Intercepts In Exercises \(19-28,\) find any intercepts. \(y=x^{2}+x-2\)

Short Answer

Expert verified
The y-intercept is -2 and the x-intercepts are -2 and 1.

Step by step solution

01

Find the Y-intercept

To find the y-intercept, substitute \(x = 0\) into the equation: \(y = (0)^2 + 0 - 2 = -2\) So, the y-intercept is -2.
02

Find the X-intercepts

To find the x-intercepts, set \(y = 0\) in the equation and solve for \(x\): \[0 = x^{2} + x - 2\]. This is a quadratic equation in the form \(ax^{2} + bx + c = 0\). To solve it, one can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). After substituting \(a = 1, b = 1, c = -2\), the formula becomes \(x = \frac{-1 \pm \sqrt{(1)^{2}-4(1)(-2)}}{2(1)}\), which gives two solutions.
03

Calculate the X-intercepts

Calculate the two solutions separately: 1. \(x = \frac{-1 + \sqrt{(1)+8}}{2} = 1\), 2. \(x = \frac{-1 - \sqrt{(1)+8}}{2} = -2\). So, the x-intercepts are \(x = 1\) and \(x = -2\)

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.

Y-intercept
The y-intercept is the point where a graph crosses the y-axis. In mathematical terms, it is found by setting the value of x to zero and solving for y.
For the quadratic equation given, \(y = x^2 + x - 2\), substitute \(x = 0\). This simplifies to \(y = 0^2 + 0 - 2 = -2\).
Therefore, the y-intercept is -2.
This means that the graph will cross the y-axis at the point (0, -2).
Laying this out clearly shows how easy it can be to find where a graph touches the y-axis, which is invaluable for sketching curves.
X-intercepts
Unlike the y-intercept, x-intercepts are the points where the graph crosses the x-axis. To find these points, we need to set y to zero in the equation and solve for x.
For the equation \(y = x^2 + x - 2\), set \(y = 0\), giving us \(x^2 + x - 2 = 0\).
These are the x-intercepts. Solving this quadratic equation will yield the x-values where the graph touches or crosses the x-axis.
Finding these points helps understand where the graph lies in relation to the x-axis and can show key points of contact or change in direction.
Quadratic Equation
Quadratic equations are polynomial equations of the second degree, typically written in the form \(ax^2 + bx + c = 0\).
The equation in the exercise is in this format, with \(a = 1\), \(b = 1\), and \(c = -2\).
Quadratic equations characteristically form parabolas when graphed. Depending on the value and sign of \(a\), these parabolas can open upwards or downwards.
Understanding quadratic equations is key to solving for x-intercepts, y-intercepts, and analyzing the behavior of parabolic graphs in various contexts.
Quadratic Formula
The quadratic formula is a solution formula for quadratic equations, given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
It provides a method to find the roots of any quadratic equation by plugging in the values of \(a\), \(b\), and \(c\).
For the equation \(x^2 + x - 2 = 0\), applying this formula gives \(x = \frac{-1 \pm \sqrt{1^2 + 8}}{2}\), solving which gives two solutions: \(x = 1\) and \(x = -2\).
The quadratic formula is a reliable method to find where the parabola intersects the x-axis, thus identifying the x-intercepts with ease.

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

Finding Composite Functions In Exercises \(63-66,\) find the composite functions \(f^{\circ} g\) and \(g \circ f\) . Find the domain of each composite function. Are the two composite functions equal? $$\begin{array}{l}{f(x)=x^{2}} \\ {g(x)=\sqrt{x}}\end{array}$$

Sketching the Graph of a Trigonometric Function In Exercises \(55-66,\) sketch the graph of the function. $$y=\sin (x+\pi)$$

Automobile Aerodynamics The horsepower \(H\) required to overcome wind drag on a certain automobile is $$\begin{array}{l}{H(x)=0.00004636 x^{3}} \\ {\text { where } x \text { is the speed }} \\ {\text { of the car in miles }} \\\ {\text { per hour. }} \\ {\text { (a) Use a graphing }} \\ {\text { atility to graph } H} \\ {\text { Find } H(x / 1.6) .}\end{array}$$

Deciding Whether an Equation Is a Function In Exercises \(43-46,\) determine whether \(y\) is a function of \(x\). $$x^{2}+y=16$$

Writing Functions In Exercises 79-82, write an equation for a function that has the given graph. $$$$Line segment connecting \((-2,4)\) and \((0,-6)\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

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.