/*! 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 1 The graph of each equation is a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 1

The graph of each equation is a parabola. Determine whether the parabola opens upward, downward, to the left, or to the right. Do not graph. $$y=x^{2}-7 x+5$$

Short Answer

Expert verified
The parabola opens upwards.

Step by step solution

01

Identify the Standard Form

First, identify the standard form of the parabola. The given equation is \( y = x^2 - 7x + 5 \). This equation is in the form \( y = ax^2 + bx + c \), which is the standard form for a parabola opening either upwards or downwards.
02

Determine the Direction of Opening

Check the coefficient \( a \) in the standard form \( y = ax^2 + bx + c \). Here, \( a = 1 \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), the parabola opens downwards. Since \( a = 1 \), which is positive, the parabola opens upwards.

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 Equation
A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). This equation forms the basis for understanding parabolas in algebra.

Quadratic equations can have several characteristics, such as:
  • The variable \( x \) is squared, which defines the quadratic nature of the equation.
  • The equation can be solved using different methods like factoring, completing the square, or the quadratic formula.
  • Solutions to quadratic equations are the x-values where the graph of the equation (parabola) intersects the x-axis, often referred to as roots or solutions.

Solving these equations is crucial in many areas of mathematics and applied sciences as they model a wide variety of physical phenomena.
Parabola Opening Direction
The direction in which a parabola opens is determined by the coefficient of the squared term in its equation. For quadratic equations in the standard form \( y = ax^2 + bx + c \):

  • If the coefficient \( a \) of \( x^2 \) is positive, the parabola opens upward.
  • If the coefficient \( a \) is negative, the parabola opens downward.

This behavior is because the sign of \( a \) determines how the values of \( y \) behave as \( x \) becomes very large or very small.
Upward-opening parabolas have a minimum point called the vertex, whereas downward-opening parabolas have a vertex that represents their maximum point. Comprehending the opening direction is helpful for sketching the graph without plotting multiple points.
Standard Form of a Parabola
The standard form of a parabola concerning quadratic equations is expressed as \( y = ax^2 + bx + c \). This form is a simple representation that helps identify many characteristics of the parabola effortlessly.

Characteristics of this form of a parabola include:
  • \( a \): Coefficient that defines the direction of opening and width of the parabola.
  • \( b \): Influences the position of the vertex along the x-axis.
  • \( c \): Represents the y-intercept, the point where the parabola crosses the y-axis.

Recognizing the standard form allows for easy transitions to other forms such as vertex form, which can further illuminate the properties of a parabola.

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

A rectangular holding pen for cattle is to be designed so that its perimeter is 92 feet and its area is 525 feet. Find the dimensions of the holding pen.

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$x^{2}+y^{2}+6 x+10 y-2=0$$

We know that \(x^{2}+y^{2}=25\) is the equation of a circle. Rewrite the equation so that the right side is equal to \(1 .\) Which type of conic section does this equation form resemble? In fact, the circle is a special case of this type of conic section. Describe the conditions under which this type of conic section is a circle.

Find each function value if \(f(x)=3 x^{2}-2 .\) See Section 3.2 $$ f(-1) $$

Rationalize each denominator and simplify if possible. $$\frac{\sqrt{5}}{\sqrt{8}}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

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.