/*! 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 58 Find the \(x\) - and \(y\) -inte... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 58

Find the \(x\) - and \(y\) -intercepts. $$ x^{2}=-y+16 $$

Short Answer

Expert verified
The x-intercepts are (4, 0) and (-4, 0), and the y-intercept is (0, 16).

Step by step solution

01

Set up the equation for the y-intercept

To find the y-intercept, set x to 0 in the equation. This will give the point where the graph intersects the y-axis.
02

Solve for y

Substitute 0 for x in the equation: \(0^2 = -y + 16\) This simplifies to: \(-y + 16 = 0\) Next, solve for y: \( y = 16\) So, the y-intercept is (0, 16).
03

Set up the equation for the x-intercepts

To find the x-intercepts, set y to 0 in the equation. This will give the points where the graph intersects the x-axis.
04

Solve for x

Substitute 0 for y in the equation: \( x^2 = -0 + 16\) This simplifies to: \( x^2 = 16\) Now, take the square root of both sides: \( x = \pm \sqrt{16}\) So, the x-intercepts are (4, 0) and (-4, 0).

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.

quadratic equations
Quadratic equations are a type of polynomial equation in which the highest exponent of the variable is 2. These equations take the form: ax^2 + bx + c = 0 where a, b, and c are constants. In the exercise, the equation is not in the standard form, but we can still identify it as quadratic because the highest power of x is 2. To solve a quadratic equation, you can use several techniques:
  • Factoring
  • Completing the Square
  • Quadratic Formula: (-b 卤 鈭�(b^2 - 4ac)) / 2a
    • In this particular example, we solved for the x- and y-intercepts by substituting 0 for y and x respectively. The solutions for x involve taking square roots, which is a direct method applicable here.
    coordinate geometry
    Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. This allows us to describe shapes, lines, and curves in a mathematical manner. By plotting points (x, y) on a graph:
    • We can visualize the solutions of equations.
    • It helps in understanding the relationship between algebra and geometry.
    • It clarifies the meaning of finding intercepts, slopes, and other geometric properties.
    In the provided exercise, the graph of the curved equation x^2 = -y + 16 intersects the x-axis and the y-axis. By substituting specific values for x and y, we can find these intersection points, known as intercepts. Coordinating geometry helps us visualize where the graph crosses the axes.
    intercepts
    Intercepts are points where a graph intersects the coordinate axes. There are two main types: x-intercepts and y-intercepts:
    • X-intercepts: Points where the graph intersects the x-axis. These are found by setting y=0 and solving for x. In the exercise, this gave the points (4, 0) and (-4, 0).

    • Y-intercepts: Points where the graph intersects the y-axis. These are found by setting x=0 and solving for y. This process resulted in the y-intercept (0, 16) in the exercise.
    These intercepts provide crucial information about the graph's shape and position. Knowing where the graph crosses the axes is essential for understanding the behavior and characteristics of quadratic equations and other polynomial functions.

    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

    Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(6) $$

    Graph the function. $$ h(x)=\operatorname{int}(2 x) $$

    Given \(f(x)=\sqrt{x+3}\) a. Find the difference quotient. b. Rationalize the numerator of the expression in part (a) and simplify. c. Evaluate the expression in part (b) for \(h=0\).

    From an equation in \(x\) and \(y\), explain how to determine whether the graph of the equation is symmetric with respect to the \(x\) -axis, \(y\) -axis, or origin.

    Graph the function. $$ k(x)=\operatorname{int}\left(\frac{1}{2} x\right) $$
    See all solutions

    Recommended explanations on Math Textbooks

    Calculus

    Read Explanation

    Decision Maths

    Read Explanation

    Logic and Functions

    Read Explanation

    Statistics

    Read Explanation

    Mechanics Maths

    Read Explanation

    Probability and 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.