/*! 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 8 For \(f(x)=-x^{2}+6 x-7,\) the v... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 8

For \(f(x)=-x^{2}+6 x-7,\) the value of \(-\frac{b}{2 a}\) is \(3 .\) Find the \(y\) -coordinate of the vertex of the graph of this function.

Short Answer

Expert verified
The \(y\)-coordinate of the vertex is 2.

Step by step solution

01

Identify the coefficients

Given the quadratic function \(f(x) = -x^2 + 6x - 7\), the coefficients are \(a = -1\), \(b = 6\), and \(c = -7\).
02

Recall the vertex formula

The formula for the \(x\)-coordinate of the vertex in a quadratic equation \(f(x) = ax^2 + bx + c\) is \(x = -\frac{b}{2a}\). We are given that this value is 3.
03

Verify the given \(x\)-coordinate formula

Substitute \(a = -1\) and \(b = 6\) into the formula: \(x = -\frac{6}{2(-1)} = 3\). This confirms the given \(x\)-coordinate of the vertex is correct.
04

Substitute \(x\) into the function to find \(y\)

Use the \(x\)-coordinate to find the \(y\)-coordinate of the vertex. Substitute \(x = 3\) into the function: \(f(3) = -(3)^2 + 6(3) - 7\).
05

Calculate \(f(3)\)

Calculate \(f(3)\): \(-(3)^2 + 6(3) - 7 = -9 + 18 - 7 = 2\). Therefore, the \(y\)-coordinate of the vertex is 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.

Understanding Quadratic Functions
A quadratic function is a type of polynomial function with a degree of 2, often expressed in the standard form as \( f(x) = ax^2 + bx + c \). In this form, \( a \), \( b \), and \( c \) are coefficients where \( a eq 0 \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \).
Here's what you need to know:
  • If \( a > 0 \), the parabola opens upward, forming a U-shape.
  • If \( a < 0 \), the parabola opens downward, creating an upside-down U-shape.
  • The values of \( b \) and \( c \) affect the symmetry and vertical position of the graph but do not impact the direction it opens.
Understanding the structure of a quadratic function helps in finding important points such as vertices, intercepts, and axis of symmetry, which are crucial in coordinate geometry.
Mastering the Vertex Formula
In coordinate geometry, the vertex of a quadratic function's parabola is a significant point, often considered its 'highest' or 'lowest' spot depending on its orientation. To locate the vertex, we use the vertex formula for finding the \( x \)-coordinate: \( x = -\frac{b}{2a} \).
Let's break it down:
  • The formula \( x = -\frac{b}{2a} \) helps determine the axis of symmetry of the parabola, an imaginary vertical line that divides the parabola into two mirror-image halves.
  • Once the \( x \)-coordinate is found, it's simple to determine the entire vertex by substituting this value back into the original quadratic function to find the corresponding \( y \)-coordinate.
  • This process directly informs us where the vertex lies in relation to the rest of the graph. For instance, with the function \( f(x) = -x^2 + 6x - 7 \), finding \( x = 3 \) via the formula, then substituting back to find \( y = 2 \), pinpoints the vertex at \( (3, 2) \).
The vertex provides a peak or a trough on the graph, aiding in understanding the quadratic's real-world dynamics.
Exploring Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach allows for precise calculations and representations of shapes through algebraic equations. When dealing with quadratics:
  • The coordinates \((x, y)\) on a plane are used to portray relationships derived from the equation of the function.
  • The vertex found via quadratic exploration serves as a critical point where the curve changes direction, helping in graphing or solving optimization problems.
  • Understanding intercepts, such as where the graph crosses the \( x \) or \( y \) axis, can also be achieved through coordinate geometry techniques. These points are essential for sketching the function with accuracy.
This intersection between algebra and geometry provides a comprehensive toolkit for deeper mathematical understanding and problem-solving.

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

Solve: \(\sqrt{x^{2}+10}=4 \sqrt{x}\)

On an exam, a student was asked to solve the equation \(-4 w^{2}-6 w-1=0 .\) Her first step was to multiply both sides of the equation by \(-1 .\) She then used the quadratic formula to solve \(4 w^{2}+6 w+1=0\) instead. Is this a valid approach? Explain.

All of the equations we have solved so far have had rational-number coefficients. However, the quadratic formula can be used to solve quadratic equations with irrational or even imaginary coefficients. Solve each equation. $$ \sqrt{2} x^{2}+x-\sqrt{2}=0 $$

Water Usage. The height (in feet) of the water level in a reservoir over a 1 -year period is modeled by the function \(H(t)=3.3(t-9)^{2}+14\) where \(t=1\) represents January, \(t=2\) represents February, and so on. How low did the water level get that year, and when did it reach the low mark?

An architect needs to determine the height \(h\) of the window shown in the illustration. The radius \(r,\) the width \(w,\) and the height \(h\) of the circular- shaped window are related by the formula \(r=\frac{4 h^{2}+w^{2}}{8 h} .\) If \(w\) is to be 34 inches and \(r\) is to be 18 inches, find \(h\) to the nearest tenth of an inch. (IMAGE CAN'T COPY)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.