/*! 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 31 Find the vertex of the graph of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 31

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples I through \(4 .\) (IMAGE CANNOT COPY) \(f(x)=x^{2}+1\)

Short Answer

Expert verified
Vertex is at (0, 1); the graph opens upward with no x-intercepts and a y-intercept at (0, 1).

Step by step solution

01

Identify the Quadratic Function

The given quadratic function is \( f(x) = x^2 + 1 \). This is a standard form quadratic equation where \( a = 1 \), \( b = 0 \), and \( c = 1 \).
02

Find the Vertex

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Since our quadratic is already in standard form, use the formula \( h = -\frac{b}{2a} \). Here, \( b = 0 \) and \( a = 1 \), so \( h = -\frac{0}{2 \times 1} = 0 \). Substitute \( x = 0 \) in \( f(x) \) to find \( k \): \( k = f(0) = 0^2 + 1 = 1 \). Thus, the vertex is \( (0, 1) \).
03

Determine the Direction of Opening

The sign of coefficient \( a \) determines the direction of the parabola. Since \( a = 1 \), which is positive, the parabola opens upward.
04

Find the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). Calculate \( f(0) = 0^2 + 1 = 1 \). Therefore, the y-intercept is at \( (0, 1) \).
05

Identify Any X-Intercepts

X-intercepts occur where \( f(x) = 0 \). Solve \( x^2 + 1 = 0 \). Rearranging gives \( x^2 = -1 \), which has no real solutions because the square root of a negative number is not real. Therefore, there are no x-intercepts.
06

Sketch the Graph

To sketch the graph, plot the vertex \((0, 1)\) and understand that the parabola opens upward. The graph does not intersect the x-axis and crosses the y-axis at \( (0, 1) \), the vertex point. The graph is symmetric with respect to the y-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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Function
A quadratic function represents a mathematical expression of the form \( f(x) = ax^2 + bx + c \). Regardless of the complexity, this function will always graph out as a parabola. The parameter \( a \) influences the shape and direction of the parabola opening.
- If \( a > 0 \), the parabola opens upwards, resembling a 'U' shape. - If \( a < 0 \), it opens downwards, similar to an upside-down 'U'.
In our problem, the function is \( f(x) = x^2 + 1 \) which is already a simple example of a quadratic function, with \( a = 1 \), meaning the parabola opens upward.
Standard Form
The standard form of a quadratic function is written as \( ax^2 + bx + c \). It provides a simple way to identify the key components of a parabola, namely:
- \( a \) which influences the direction and width of the parabola. - \( b \) which affects the horizontal placement and slope of the parabola's line of symmetry. - \( c \) which is the value of the y-intercept.
In our case, the equation \( f(x) = x^2 + 1 \) matches the standard form with \( a = 1 \), \( b = 0 \), and \( c = 1 \). This information lets us quickly find features like the vertex and intercepts.
Y-Intercept
The y-intercept of a graph is where it crosses the y-axis, which happens when \( x = 0 \). In a quadratic function like \( ax^2 + bx + c \), the y-intercept is simply \( c \) since that results from substituting \( x = 0 \) into \( f(x) \).
For our function \( f(x) = x^2 + 1 \), substituting \( x = 0 \) gives:
\[ f(0) = 0^2 + 1 = 1 \]
Thus, the y-intercept is the point \((0, 1)\). This tells you exactly where the parabola starts its journey along the y-axis.
X-Intercept
X-intercepts are the points where the graph of a quadratic function crosses the x-axis. These points represent the values of \( x \) where \( f(x) = 0 \). To find x-intercepts, solve the equation \( ax^2 + bx + c = 0 \).
For the function \( f(x) = x^2 + 1 \), setting it to zero gives:
\[ x^2 + 1 = 0 \]
Solving for \( x \) gets you \( x^2 = -1 \), which doesn鈥檛 have real solutions since the square root of a negative number is not real. Therefore, our parabola doesn鈥檛 cross the x-axis and has no x-intercepts.
This concept explains why sometimes, for certain functions, portions of a parabola may "float" above the x-axis without touching or crossing it.

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

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ H(x)=\frac{3}{4} x^{2}-2 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=-(x-2)^{2} $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ g(x)=-(x+6)^{2} $$

Use the quadratic formula to solve each quadratic equation. $$ x^{2}+\sqrt{2} x+1=0 $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples I through 5 . $$ h(x)=x^{2}+5 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.