/*! 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 70 Solve each formula for the speci... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 70

Solve each formula for the specified variable.} \(y=a(x-h)^{2}+k\) for \(a\) (Mathematics)

Short Answer

Expert verified
\(a = \frac{y-k}{(x-h)^{2}}\)

Step by step solution

01

Isolate the Quadratic Expression

Given the function \(y=a(x-h)^{2}+k\), first isolate the term involving \(a\). Subtract \(k\) from both sides to begin moving terms that don't involve \(a\): \[ y - k = a(x-h)^{2} \] This rearranges the equation to focus on the term with \(a\).
02

Solve for the Variable \(a\)

With the equation \(y-k = a(x-h)^{2}\), divide both sides by \((x-h)^{2}\) to solve for \(a\):\[ a = \frac{y-k}{(x-h)^{2}} \]This isolates \(a\) as the subject of the formula.

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 Expressions
A quadratic expression is a polynomial of degree two, which means it involves a variable raised to the power of two. In this exercise, the quadratic expression is in the form \(a(x-h)^{2}\), which is part of the function \(y=a(x-h)^{2}+k\). This specific form is known as the vertex form of a quadratic equation, and it is beneficial in identifying the properties of a quadratic, such as its vertex.
  • The quadratic term is \(x-h\), which indicates a horizontal shift.
  • The square \( (x-h)^{2}\) means the parabola opens either upwards or downwards, depending on the sign and value of \(a\).
  • The constant \(k\) shows the vertical shift, moving the entire graph up or down.
Understanding this structure makes it easier to manipulate and solve equations involving quadratics.
Formula Manipulation
Formula manipulation involves rearranging the components of an equation to solve for a specific variable or to simplify the equation. In our exercise, we started with the equation \(y=a(x-h)^{2}+k\) and manipulated it to solve for \(a\). Manipulating formulas is a critical skill in algebra that enables students to:
  • Understand relationships between variables by rearranging terms.
  • Isolate specific variables to make them the subject of the equation.
  • Simplify complex equations for better interpretation or computation.
Effective manipulation starts by recognizing the structure of the formula and knowing which mathematical operations can be applied to both sides without changing the equality. For instance, subtracting \(k\) from both sides began the process of isolating the term \(a(x-h)^{2}\).
Variable Isolation
Variable isolation is the process of rearranging an equation to make one variable the subject. This typically requires reverse operations like addition or subtraction first, followed by division or multiplication. In this problem, we were tasked with isolating \(a\) in the function \(y=a(x-h)^{2}+k\).The isolated form of the equation provides direct information about how \(a\) varies with changes in other variables. Specifically, we:
  • First, subtracted \(k\) from both sides to bring \(a(x-h)^{2}\) away from other terms.
  • Next, divided both sides by \((x-h)^{2}\) to completely isolate \(a\).
  • The final isolated expression, \ a = \frac{y-k}{(x-h)^{2}} \, clearly indicates \(a\)'s dependence on \(y\), \(x\), \(h\), and \(k\).
Mastering variable isolation allows for solving equations in various forms, giving us a clearer understanding of mathematical relationships.

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

Passing through \(\left(\frac{3}{4}, \frac{1}{4}\right)\) and perpendicular to the line passing through \((-3,-5)\) and \((-4,0)\)

Match each equation with the graph that it most closely resembles. $$y=-3 x-6$$

In the late \(1920 \mathrm{s}\), the famous observational astronomer Edwin P. Hubble ( \(1889-1953\) ) determined the distances to several galaxies and the velocities at which they were receding from Earth. Four galaxies with their distances in light-years and velocities in miles per second are listed in the table at the top of the next column. $$\begin{array}{|l|c|c|}\quad\quad\text { Galaxy } & \text { Distance } & \text { Velocity } \\\\\hline \text { Virgo } & 50 & 990 \\\\\text { Ursa Minor } & 650 & 9,300 \\\\\text { Corona Borealis } & 950 & 15,000 \\\\\text { Bootes } & 1,700 & 25,000\end{array}$$ (a) Let \(x\) represent velocity and \(y\) represent distance. Find the equation of the least-squares regression line that models the data. (b) If the galaxy Hydra is receding at a speed of \(37,000\) miles per second, estimate its distance from Earth.

Solve each problem. Two ships leave the same harbor at the same time. The first ship heads north at 20 mph and the second ship heads west at 15 mph. (a) Draw a sketch depicting their positions after t hours. (b) Write an expression that gives the distance between the ships after t hours.

Sales Tax If the sales tax rate is \(7.5 \%,\) write a function \(f\) that calculates the sales tax on a purchase of \(x\) dollars. What is the sales tax on a purchase of \(\$ 86 ?\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.