/*! 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 67 Find the equation of the parabol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 67

Find the equation of the parabola (with vertical axis that passes through the data points shown or specified. Check your answer. $$(1.5,6.25),(0,-2),(-1.5,3.25)$$

Short Answer

Expert verified
The equation of the parabola is \(y = 3x^2 + x - 2\).

Step by step solution

01

General Form of a Parabola

The equation of a parabola with a vertical axis is typically given by \(y = ax^2 + bx + c\). We will use this form to find constants \(a\), \(b\), and \(c\) that describe the parabola passing through the given points.
02

Plug in the First Point

Substitute \((x, y) = (1.5, 6.25)\) into the equation: \[6.25 = a(1.5)^2 + b(1.5) + c\] which simplifies to: \[6.25 = 2.25a + 1.5b + c\].
03

Plug in the Second Point

Substitute \((x, y) = (0, -2)\) into the equation: \[-2 = a(0)^2 + b(0) + c\] which simplifies to: \[-2 = c\].
04

Plug in the Third Point

Substitute \((x, y) = (-1.5, 3.25)\) into the equation: \[3.25 = a(-1.5)^2 + b(-1.5) + c\] which simplifies to: \[3.25 = 2.25a - 1.5b + c\].
05

Solve for 'a' and 'b' using First and Third Points

With \(c = -2\), substitute into equations obtained from steps 2 and 4: \[6.25 = 2.25a + 1.5b - 2\]\[3.25 = 2.25a - 1.5b - 2\]Simplify both to get:\[8.25 = 2.25a + 1.5b\]\[5.25 = 2.25a - 1.5b\].
06

Eliminate 'a' by Adding Equations

Add the simplified equations from Step 5:\[8.25 + 5.25 = (2.25a + 1.5b) + (2.25a - 1.5b)\]\[13.5 = 4.5a\]\[a = 3\].
07

Substitute to Find 'b'

Now substitute \(a = 3\) back into one of the equations:\[8.25 = 2.25(3) + 1.5b\]\[8.25 = 6.75 + 1.5b\]\[1.5 = 1.5b\]\[b = 1\].
08

Write the Final Equation

We have found \(a = 3\), \(b = 1\), and \(c = -2\). Thus, the equation of the parabola is:\[y = 3x^2 + x - 2\].

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.

Vertical axis
When we talk about a parabola with a vertical axis, it means the parabola opens either upwards or downwards. In mathematical terms, a parabola with a vertical axis can be represented by the equation: \[ y = ax^2 + bx + c \] Here, the vertical axis refers to the "y" direction, which indicates how the parabola stretches or compresses along the "y" axis.
  • The coefficient \( a \) determines the direction of the opening. If \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.
  • The vertex lies on the vertical line called the axis of symmetry, which is found using \( x = -\frac{b}{2a} \).
Recognizing the orientation of the parabola helps in predicting its behavior and plotting it accurately on a graph.
Solving systems of equations
A system of equations is a set of multiple equations that you solve simultaneously. In problems involving parabolas, solving these systems helps determine specific values for variables that describe the shape and position of a parabola. Consider this scenario when solving systems of equations with \(a\), \(b\), and \(c\) to form a parabola:
  • Start by writing equations from the given points by substituting them into the parabola equation.
  • Simplify the system of linear equations generated from each point.
  • Use algebraic methods to simultaneously solve for unknown parameters.
By using the information from the given points, this method allows us to systematically find the parameters \(a\), \(b\), and \(c\) essential for forming the correct parabola equation.
Substitution Method
The substitution method is often used to solve systems of equations, especially when a direct value is already known for one variable. In our example, after writing the equations from point substitutions, we quickly identified that \( c = -2 \) by plugging in the point \((0, -2)\) into the parabola equation. Here's how it works:
  • Solve one of the equations for one variable when its value is readily apparent. Here, we found \(c\) using the point where \(x = 0\).
  • Substitute this value into the other equations. This reduces the number of unknowns and simplifies the problem.
  • Continue this process until all variables are solved.
Using substitution simplifies the solving process by focusing on one unknown at a time, making complex systems much more manageable.
Quadratic Function
A quadratic function is defined as a second-degree polynomial function of the form \( y = ax^2 + bx + c \), where \( a eq 0 \). The graph of a quadratic function is a parabola. Understanding the characteristics of quadratic functions is crucial for analyzing their graphs and properties. Some key features of a quadratic function include:
  • The parabola can open upwards or downwards depending on the sign of the coefficient \( a \).
  • The vertex provides the highest or lowest point on the graph, depending on the parabola's orientation.
  • The quadratic formula or factoring can be used to find the roots or x-intercepts: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Recognizing these features allows students to solve for and graph quadratic functions accurately, aiding in the comprehensive understanding of their behavior.

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

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$x+y \geq 2$$ $$x+y \leq 6$$

Use the shading capabilities of a graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq 2^{x}\\\&y \leq 8\end{aligned}$$

Fill in each blank with the appropriate response. The graph of the system,$$y>x^{2}+2$$ $$x^{2}+y^{2}<16\( $$y<7$$ consists of all points \)\frac{ }{\text { (above/below) }}\( the parabola given by \)y=x^{2}+2, \frac{ }{(\text { insideloutside })}\( the circle \)x^{2}+y^{2}=16,\( and \)\frac{ }{\text { (above/below) }}\( the line \)y=7$.

Which one of the following is a description of the graph of the inequality \((x-5)^{2}+(y-2)^{2}<4 ?\) A. The region inside a circle with center \((-5,-2)\) and radius 2 B. The region inside a circle with center \((5,2)\) and radius 2 C. The region inside a circle with center \((-5,-2)\) and radius 4 D. The region outside a circle with center \((5,2)\) and radius 4

Solve each system by using the matrix inverse method. $$\begin{aligned} x+3 y &=-12 \\ 2 x-y &=11 \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.