/*! 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 32 Graph the function, label the ve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 32

Graph the function, label the vertex, and draw the axis of symmetry. $$ f(x)=-3(x+7)^{2} $$

Short Answer

Expert verified
Vertex: (-7, 0); Axis of Symmetry: x = -7

Step by step solution

01

Identify the Parabola Form

The given function is already in vertex form, which is \[f(x) = a(x-h)^2 + k\]where \(a = -3\), \(h = -7\), and \(k = 0\). This tells us our parabola opens downwards due to the negative leading coefficient.
02

Determine the Vertex

The vertex of the parabola in vertex form \(f(x) = a(x-h)^2 + k\) is given by the point \((h, k)\). So, the vertex here is \((-7, 0)\).
03

Draw the Axis of Symmetry

The axis of symmetry for a parabolic function of the form \(f(x) = a(x-h)^2 + k\) is the vertical line \(x = h\). Therefore, for the given function, the axis of symmetry is \(x = -7\). Draw a vertical line through \(x = -7\).
04

Plot the Vertex

On a coordinate plane, plot the vertex at the point \((-7, 0)\).
05

Sketch the Parabola

Since \(a = -3\) is negative, the parabola opens downwards. From the vertex \((-7, 0)\), draw a symmetric U-shaped curve opening down.

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 parabola
Understanding the vertex form of a parabola is key to graphing quadratic functions. The vertex form is given by \(f(x) = a(x-h)^2 + k\). In this form, \(h\) and \(k\) are the coordinates of the vertex of the parabola. The value \(a\) determines the direction and the width of the parabola:
  • If \(a > 0\), the parabola opens upwards.
  • If \(a < 0\), the parabola opens downwards.
  • The larger the absolute value of \(a\), the narrower the parabola.
To graph the given function \(f(x)=-3(x+7)^2\), we see it is already in its vertex form. Here, \(a=-3\), \(h=-7\), and \(k=0\), so the vertex is at \((-7,0)\). The negative \(a\) tells us the parabola will open downwards.
axis of symmetry
The axis of symmetry is a vertical line that runs through the vertex of the parabola, dividing it into two mirror-image halves. For the vertex form \(f(x) = a(x-h)^2 + k\), the axis of symmetry has the equation \(x = h\). Because this line is vertical, it is expressed as \(x = \text{constant}\).
In the given function \(f(x)=-3(x+7)^2\), \(h = -7\). Therefore, the axis of symmetry is the line \(x = -7\). This vertical line helps in plotting symmetrical points on either side of the parabola.
plotting parabolas
To graph a parabola, start by plotting the vertex, which is the most central point. Here, for \(f(x)=-3(x+7)^2\), the vertex is \(-7,0\). Mark this point on the graph.
Next, draw the axis of symmetry \(x=-7\) as a dashed vertical line. This helps in creating a balanced graph.
From the vertex, plot additional points by choosing \(x\)-values on either side, calculating their corresponding \(y\)-values, and ensuring they mirror each other across the axis of symmetry. Finally, draw a smooth, U-shaped curve through these points to complete the parabola. Remember, since \(a\) is negative, this parabola will open downwards.
understanding vertical shifts
Vertical shifts in the vertex form \(f(x) = a(x-h)^2 + k\) can change the position of the parabola along the y-axis. The value of \(k\) determines how far and in which direction the graph shifts vertically:
  • If \(k > 0\), the parabola shifts up.
  • If \(k < 0\), the parabola shifts down.
In our function \(f(x)=-3(x+7)^2\), \(k = 0\), indicating no vertical shift. The vertex remains at the same y-coordinate as determined by \(h\) and \(k\). Recognizing these shifts helps in sketching the graph accurately.

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

Abby rows \(10 \mathrm{km}\) upstream and \(10 \mathrm{km}\) back in a total time of 3 hr. The speed of the river is \(5 \mathrm{km} / \mathrm{h} .\) Find Abby's speed in still water.

On a sales trip, Samir drives the \(600 \mathrm{mi}\) to Richmond averaging a certain speed. The return trip is made at an average speed that is 10 mph slower. Total time for the round trip is 22 hr. Find Samir's average speed on each part of the trip.

Height of a Thrown Object. The function $$S(t)=-16 t^{2}+32 t+1920$$ gives the height \(S\), in feet, of an object thrown from a cliff that is 1920 ft high. Here \(t\) is the time, in seconds, that the object is in the air. a) For what times does the height exceed 1920 ft ? b) For what times is the height less than 640 ft?

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(c=\sqrt{g H},\) for \(H\) (Velocity of ocean wave)

The NBA's LeBron James has a vertical leap of 44 in. What is his hang time? (Use \(V=48 T^{2}\) ).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.