/*! 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 12 For each part, draw a rough sket... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 12

For each part, draw a rough sketch of a graph of a function of the type \(f(x)=a x^{2}+b x+c\) a. Where \(a>0, c>0,\) and the function has no real zeros. b. Where \(a<0, c>0,\) and the function has two real zeros. c. Where \(a \geq 0\) and the function has one real zero.

Short Answer

Expert verified
a. Upward parabola above x-axis, no real zeros. b. Downward parabola crossing x-axis twice, two real zeros. c. Parabola or line touching x-axis at one point, one real zero.

Step by step solution

01

Understanding Quadratic Functions

The general form of a quadratic function is given by \(f(x)=a x^{2}+b x+c\). In this type of function, the shape of the graph is a parabola.
02

Part a: Identify Conditions

For part a: \(a>0\), \(c>0\), and the function has no real zeros. This implies the parabola opens upwards (\(a>0\)), is above the x-axis (\(c>0\)), and the discriminant (\(b^2 - 4ac < 0\)) is less than zero.
03

Part a: Graph Sketch

Sketch the parabola opening upwards and positioned above the x-axis, with no intersection points. This indicates no real zeros.
04

Part b: Identify Conditions

For part b: \(a<0\), \(c>0\), and the function has two real zeros. This implies the parabola opens downwards (\(a<0\)), intersects the y-axis above zero (\(c>0\)), and the discriminant (\(b^2 - 4ac > 0\)) is greater than zero.
05

Part b: Graph Sketch

Sketch the parabola opening downwards and crossing the x-axis at two points. This indicates two real zeros.
06

Part c: Identify Conditions

For part c: \(a \geq 0\) and the function has one real zero. This implies the parabola could open upwards (\(a>0\)) or be flat (\(a=0\)), and the vertex touches the x-axis (\(b^2 - 4ac = 0\)).
07

Part c: Graph Sketch

Sketch the parabola with the vertex touching the x-axis; it could be a parabola opening upwards or a single point if the function is linear.

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.

Parabolas
A parabola is the graph of a quadratic function. Quadratic functions have the form \( f(x) = ax^2 + bx + c \). Parabolas have unique features:
  • The direction they open depends on the sign of \(a\).
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), the parabola opens downwards.
The vertex is the highest or lowest point on the parabola:
  • For an upward-opening parabola, the vertex is the lowest point.
  • For a downward-opening parabola, the vertex is the highest point.
Parabolas mirror along their vertical line (axis of symmetry).Understanding the basics of parabolas helps you identify how different quadratic functions behave.
Discriminant
The discriminant of a quadratic function is found in the formula:\[ \text{Discriminant} = b^2 - 4ac \].The discriminant helps determine the nature of the roots (zeroes) of the quadratic equation \(ax^2 + bx + c = 0\).
  • If the discriminant is greater than zero (\(b^2 - 4ac > 0\)), there are two distinct real zeros.
  • If it is equal to zero (\(b^2 - 4ac = 0\)), there is exactly one real zero.
  • If it is less than zero (\(b^2 - 4ac < 0\)), there are no real zeros.
The discriminant provides crucial information about how the parabola interacts with the x-axis.
Graph Sketching
When sketching a quadratic graph, consider key aspects such as direction, vertex, axis of symmetry, and x-intercepts:
  • Direction: Check whether \(a\) is positive or negative to determine if the graph opens upwards or downwards.
  • Vertex: Calculate the vertex using \(x = -\frac{b}{2a}\) and then find \(f(x)\).
  • Axis of Symmetry: This is the vertical line through the vertex, given by \(x = -\frac{b}{2a}\).
  • X-Intercepts: Solve the quadratic equation \(ax^2 + bx + c = 0\) using the discriminant to find the real zeros.
Use this ordered approach to make your graph sketch accurate and easy to interpret.
Real Zeros
Real zeros (or roots) are where the graph of the quadratic function intersects the x-axis.To find them, solve for \(x\) in the equation \(ax^2 + bx + c = 0\).
  • Two Real Zeros: When \(b^2 - 4ac > 0\), the parabola intersects the x-axis at two points.
  • One Real Zero: When \(b^2 - 4ac = 0\), the parabola touches the x-axis at one point (exactly at its vertex).
  • No Real Zeros: When \(b^2 - 4ac < 0\), the graph does not intersect the x-axis at any point.
Knowing how to determine the number of real zeros helps you accurately complete and analyze quadratic graphs.

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

For each part construct a function that satisfies the given conditions. a. Has a constant rate of increase of $$\$ 15,000 /$$ year b. Is a quadratic that opens upward and has a vertex at (1,-4) c. Is a quadratic that opens downward and the vertex is on the \(x\) -axis d. Is a quadratic with a minimum at the point (10,50) and a stretch factor of 3 e. Is a quadratic with a vertical intercept of (0,3) that is also the vertex

For each of the following quadratic functions, find the vertex \((h, k)\) and determine if it represents the maximum or minimum of the function. a. \(f(x)=-2(x-3)^{2}+5\) c. \(f(x)=-5(x+4)^{2}-7\) b. \(f(x)=1.6(x+1)^{2}+8\) d. \(f(x)=8(x-2)^{2}-6\)

(Graphing program required for part (c).) The rational function \(g(x)=\frac{4 x-11}{x-3}\) can be decomposed into a sum by using the following method: a. Use the preceding method to decompose \(g(x)=\frac{5 x+22}{x-3}\). b. Describe the transformation of the function \(f(x)=\frac{1}{x}\) into \(g(x)=\frac{5 x+22}{x-3}\) c. Using technology, plot the graphs of \(f(x)\) and \(g(x)\) to verify that the transformation described in part (b) is correct.

a. Given \(f(x)=\ln x,\) describe the transformations that created \(g(x)=3 f(x+2)-4\). Find \(g(x)\). b. Use your knowledge of properties of logarithms to find any vertical and horizontal intercepts for the function \(g(x)\).

a. Find the equation of the parabola with a vertex of (2,4) that passes through the point (1,7) . b. Construct two different quadratic functions both with a vertex at (2,-3) such that the graph of one function is concave up and the graph of the other function is concave down. c. Find two different equations of a parabola that passes through the points (-2,5) and (4,5) and that opens downward. (Hint: Find the axis of symmetry.)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

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.