/*! 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 55 Graph the function. Label the \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 55

Graph the function. Label the \(x\)-intercept(s) and the \(y\)-intercept. \(g(x)=(x-2)^2-4\)

Short Answer

Expert verified
The \(y\)-intercept is at the point (0,4) and the \(x\)-intercepts are at the points (4,0) and (0,0). The graph is a parabola shifted right 2 units and down 4 units from the origin which opens upwards

Step by step solution

01

Finding the \(y\)-intercept

To find the \(y\)-intercept, set \(x\) to 0 in the equation. Hence, \(g(0) = (0-2)^2-4 = 4\). Thus, the \(y\)-intercept is (0,4)
02

Finding the \(x\)-intercepts

To find the \(x\)-intercepts, set the whole equation to 0 and solve for \(x\). Hence, \( 0 = (x-2)^2-4 \) which simplifies to \( (x-2)^2 = 4 \). Taking square root on both sides gives \(x-2 = \pm 2 \), and after simplifying, we get \(x = 4,0\) Thus, the \(x\)-intercepts are (4,0) and (0,0)
03

Plotting the Function

Plot the point (0,4) which is the \(y\)-intercept, and points (4,0) and (0,0) which are the \(x\)-intercepts on the graph. You will realize that the graph is a parabola that opens upwards. Sketch the curve of the parabola accordingly

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.

Graphing Parabolas
Parabolas are the graph representations of quadratic functions, meaning any function that can be written in the form of \( ax^2 + bx + c \). In this exercise, we are dealing specifically with the function \( g(x) = (x-2)^2 - 4 \). This is in vertex form, which is generally \( a(x-h)^2 + k \). Understanding the vertex form allows you to easily identify the vertex of the parabola, which in this case is at the point \((h, k) = (2, -4)\).

Here are basic characteristics of a parabola that are helpful when graphing:
  • The direction it opens: If the coefficient \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards.
  • The vertex: Being the highest or lowest point on the graph depending on the direction the parabola opens.
  • The axis of symmetry: A vertical line that runs through the vertex; for the above function, it is \(x = 2\).
By identifying the direction, vertex, and axis of symmetry, you set a solid foundation for graphing the function accurately.
X-Intercepts
The \(x\)-intercepts are points where the graph crosses the x-axis. At these points, the value of the function, \(g(x)\), is zero. To find the x-intercepts for \(g(x) = (x-2)^2 - 4\), we need to solve the equation \((x-2)^2 - 4 = 0\).

Simplifying this equation involves:
  • Adding 4 to both sides, resulting in \((x-2)^2 = 4\).
  • Taking the square root of both sides, yielding the solutions \(x - 2 = \pm 2\).
  • Finally, solving for \(x\): adding 2 results in \(x = 4\) and subtracting 2 gives \(x = 0\).
Thus, the \(x\)-intercepts are at the points \((4, 0)\) and \((0, 0)\). These points help in appropriately plotting the parabola on the coordinate plane, providing guidance on how the curve bends and where it influences the x-axis.
Y-Intercepts
The \(y\)-intercept of a graph is the point where the graph crosses the y-axis. At this specific point, the value of \(x\) is zero. To determine the \(y\)-intercept for the quadratic function \(g(x) = (x-2)^2 - 4\), we substitute \(x = 0\) into the equation.

The calculation goes as follows:
  • Substitute 0 for \(x\): \(g(0) = (0-2)^2 - 4\).
  • Simplify the expression to \((0-2)^2 - 4 = 4 - 4\).
  • Conclude with the result: \(g(0) = 0\), hence the \(y\)-intercept is at the point \((0, 0)\).
This result shows a particular situation where the \(y\)-intercept is also one of the \(x\)-intercepts. Plotting this point on the graph provides essential information for shaping the parabola correctly.

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

In Exercises 55-62, write the quadratic function in vertex form. Then identify the vertex. \(f(x)=x^2-8 x+19\)

Write the expression as a complex number in standard form. \((3+5 i)\left(2-7 i^4\right)\)

Solve the equation by completing the square. \(5 x(x+6)=-50\)

Write the expression as a complex number in standard form. \(\left(2+4 i^5\right)+\left(1-9 i^6\right)-\left(3+i^7\right)\)

Determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation. \((x-7)^2=9\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

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.