/*! 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 58 SKETCHING GRAPHS Sketch the grap... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 58

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-4 x^{2}+32 x-20 $$

Short Answer

Expert verified
The vertex of the function \(y=-4x^2+32x-20\) is at the point (4, 4). The parabola opens downward.

Step by step solution

01

Identify the coefficients

The coefficients from the function are: \(a = -4\), \(b = 32\), and \(c = -20\).
02

Calculate the vertex

Use the formula to find the x-coordinate of the vertex \(x = -b/2a\), so \(x = -32/(2*(-4)) = 4\). To find the y-coordinate of the vertex, substitute the x-value into the function: \(y = -4*4^2 + 32*4 - 20 = 4\). Thus, the coordinates of the vertex are (4, 4).
03

Plot the vertex and draw the graph

Plot the vertex at the point (4, 4). As the coefficient of \(x^2\) is negative, the parabola opens downward. At \(x = 0\), \(y = -20\). And at \(x = 8\), the function also equals \(y = -20\). So plot these points and then sketch the graph by connecting the points. The graph is symmetric with respect to the line \(x = 4\).

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 of a Parabola
The vertex of a parabola is a crucial point, representing the peak or the lowest point of the parabola, depending on its orientation. In the equation \[y = ax^2 + bx + c\]the vertex can be found using the formula for the x-coordinate:\[x = \frac{-b}{2a}\]Substituting this x-value back into the original equation yields the y-coordinate.
  • For example, with the function \(y = -4x^2 + 32x - 20\), the coefficients are \(a = -4\), \(b = 32\), and \(c = -20\). Using the formula, we calculate:\[x = \frac{-32}{2(-4)} = 4\]
  • Substituting \(x = 4\) back into the equation gives us the y-value:\[y = -4(4)^2 + 32(4) - 20 = 4\]
  • So, the vertex of this parabola is at the point \((4, 4)\).
This vertex tells us that the parabola reaches its highest point at \((4, 4)\) since the parabola opens downwards.
Graphing Parabolas
Graphing a parabola involves plotting points on a coordinate plane and connecting them in the shape of a curve. The key points include the vertex and other crucial points like the y-intercept and additional points to define the shape.
The process includes:
  • Plotting the Vertex: Start by plotting the vertex, which in this case is \((4, 4)\).
  • Finding Intercepts: The y-intercept is where \(x = 0\). Using the equation with \(x = 0\), we find:\[y = -4(0)^2 + 32(0) - 20 = -20\]
  • Symmetry Points: Since the parabola is symmetric about its vertex, find additional points for clarity. At \(x = 8\), the y-value is equal to the y-intercept's y-value \(-20\), so we plot \((8, -20)\).
  • Drawing the Curve: Connect these key points smoothly with a curve that opens downward.
With the vertex and symmetry points plotted, sketch the parabola to reflect its accurate shape.
Axis of Symmetry
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and plays a vital role in understanding the graph鈥檚 symmetry.
The equation for the axis of symmetry is:\[x = \frac{-b}{2a}\]
  • For the function \(y = -4x^2 + 32x - 20\), the axis of symmetry is calculated to be \(x = 4\).
  • This means the line \(x = 4\) is where you could fold the parabola so each half would match the other.
  • This line helps in quickly finding points equidistant from the vertex, simplifying graphing.
Understanding this concept helps not only in graphing but also in predicting the behavior of the quadratic function.

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

In Exercises 25 and 26 , use the vertical motion model \(\boldsymbol{h}=-\mathbf{1 6 t}^{2}+\boldsymbol{v t}+\boldsymbol{s}(\mathbf{p} . \mathbf{5 3 5})\) and the following information. You and a friend are playing basketball. You can jump with an initial velocity of 12 feet per second. You need to jump 2.2 feet to dunk a basketball. Your friend can jump with an initial velocity of 14 feet per second. Your friend needs to jump 3.4 feet to dunk a basketball. Can you dunk the ball? Can your friend? Justify your answers.

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$5 a^{2}+10=20$$

Solve the inequality and graph the solution. -12<2 x-6<4

MULTI-STEP PROBLEM You are on a team that is building a roller coaster. The vertical height of the first hill of the roller coaster is supposed to be 220 feet. According to the design, the path of the first hill can be modeled by \(y=0.039 x^{2}-0.331 x+1.850,\) where \(y\) is the vertical height in feet and \(x\) is the horizontal distance in feet. The first hill can use only 75 feet of horizontal distance. a. Use the model to determine whether the first hill will reach a height of 220 feet. b. What minimum horizontal distance is needed for the first hill to reach a vertical height of 220 feet? c. Writing Can you build the first hill high enough? Explain your findings. d. CRITICAL THINKING Is the shape of the graph the same as the shape of the hill? Why or why not?

Sketch the graph of the function. Label the vertex. y=-5 x^{2}-0.5 x+0.5
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Geometry

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.