/*! 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 60 Write the vertex form of the qua... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 60

Write the vertex form of the quadratic function whose graph has vertex (-3,8) and \(y\) -intercept 5 .

Short Answer

Expert verified
\[ y = -\frac{1}{3}(x + 3)^2 + 8 \]

Step by step solution

01

- Understand the vertex form

The vertex form of a quadratic function is given by \[ y = a(x - h)^2 + k \]where \((h, k)\) is the vertex of the parabola. Here, the vertex is \((-3, 8)\).
02

- Substitute the vertex coordinates

Substitute \( h = -3 \) and \( k = 8 \) into the vertex form equation to get \[ y = a(x + 3)^2 + 8 \]
03

- Use the y-intercept to find parameter \(a\)

To find the value of \(a\), use the given y-intercept, which is 5. The y-intercept occurs when \(x = 0\). So we substitute \( x = 0 \) and \( y = 5 \) into the equation: \[ 5 = a(0 + 3)^2 + 8 \]Simplify and solve for \(a\): \[ 5 = 9a + 8 \]\[ 5 - 8 = 9a \]\[ -3 = 9a \]\[ a = -\frac{1}{3} \]
04

- Write the final vertex form

Substitute \( a = -\frac{1}{3} \) back into the equation from Step 2:\[ y = -\frac{1}{3}(x + 3)^2 + 8 \]

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.

Vertex Form of a Quadratic Function
The vertex form of a quadratic function is a special way to write quadratic functions, making it easy to see their vertex. The equation is written as: \(y = a(x - h)^2 + k\).
This method is handy because it visually tells you where the parabola's peak or low point is on a graph. Here, the vertex \((h, k) \) shows the coordinates of this point.
In our exercise, the vertex given is \((-3, 8)\). This means when we substitute in the vertex, our initial equation becomes: \[ y = a(x + 3)^2 + 8 \]. With a clear vertex, identifying other parts of the quadratic function becomes easier.
Understanding Quadratic Functions
A quadratic function is a polynomial of degree 2 that can describe various parabolic shapes, depending on its coefficients. These functions can appear in standard form as \(ax^2 + bx + c\), or, when rewritten using different methods, vertex form: \(y = a(x - h)^2 + k\).
This second format is particularly helpful because it emphasizes the vertex, offering key insights into the parabola's shape and position:
  • The coefficient 'a' influences the width and direction.
  • The vertex coordinates \((h, k)\) consist of the turning point of the curve.
Solving for Parameter a
To determine 'a' in our problem, we use another given point on the parabola鈥攖he y-intercept. The y-intercept happens when \(x = 0\), simplifying the calculation.
Our equation from the earlier steps was \(y = a(x + 3)^2 + 8\). By setting \(y = 5\) when \(x = 0\), we follow these steps:
  • Insert these values: \(5 = a(0 + 3)^2 + 8\)
  • Simplify to get: \(5 = 9a + 8\)
  • Solve for 'a' by isolating it: \(5 - 8 = 9a\), leading to \(a = -\frac{1}{3}\).
This value of 'a' helps finalize the quadratic function in vertex form.
Understanding the Y-Intercept
The y-intercept is an essential characteristic of a quadratic function, as it reveals where the graph intersects the y-axis.
For our equation \(y = -\frac{1}{3}(x + 3)^2 + 8\), the y-intercept is known to be 5 which was crucial for solving 'a'. When \(x = 0\), the equation simplifies for direct solving.
Understanding the y-intercept can also describe the graph's vertical positioning, helping predict other key points on the graph and factoring functions in various applications.

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 fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. A planet orbits a star in an elliptical orbit with the star located at one focus. The perihelion of the planet is 5 million miles. The eccentricity \(e\) of a conic section is \(e=\frac{c}{a} .\) If the eccentricity of the orbit is \(0.75,\) find the aphelion of the planet.

Find the center, transverse axis, vertices, foci, and asymptotes, Graph each equation. \(x^{2}-y^{2}-2 x-2 y-1=0\)

A homeowner is putting in a fireplace that has a 4 -inch-radius vent pipe. He needs to cut an elliptical hole in his roof to accommodate the pipe. If the pitch of his roof is \(\frac{5}{4}(\) a rise of \(5,\) run of 4\(),\) what are the dimensions of the hole?

The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadway to the arch at 5 -foot intervals along the roadway.

Analyze each equation. \(\frac{(y+2)^{2}}{16}-\frac{(x-2)^{2}}{4}=1\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.