/*! 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 12 Sketch the graph of the function... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 12

Sketch the graph of the function by first making a table of values. \(f(x)=x^{2}-4\)

Short Answer

Expert verified
The graph of \(f(x) = x^2 - 4\) is a parabola opening upwards with vertex at \((0, -4)\).

Step by step solution

01

Choose Values for x

To begin, we need to select a range of x-values to make a table. For simplicity, let's choose: \(-3, -2, -1, 0, 1, 2,\) and \(3\). You can choose more values if desired for a more detailed graph.
02

Calculate f(x) for Each x

Next, calculate \(f(x) = x^2 - 4\) for each selected \(x\) value.\(\)- For \(x = -3\), \(f(x) = (-3)^2 - 4 = 5\).\(\)- For \(x = -2\), \(f(x) = (-2)^2 - 4 = 0\).\(\)- For \(x = -1\), \(f(x) = (-1)^2 - 4 = -3\).\(\)- For \(x = 0\), \(f(x) = 0^2 - 4 = -4\).\(\)- For \(x = 1\), \(f(x) = 1^2 - 4 = -3\).\(\)- For \(x = 2\), \(f(x) = 2^2 - 4 = 0\).\(\)- For \(x = 3\), \(f(x) = 3^2 - 4 = 5\).
03

Create a Table of Values

Now, make a table in which you list each \(x\) value alongside its corresponding \(f(x)\) value from Step 2.\[\begin{array}{c|c} x & f(x) \\hline-3 & 5 \-2 & 0 \-1 & -3 \0 & -4 \1 & -3 \2 & 0 \3 & 5 \\end{array}\]
04

Plot Points on the Graph

Using the table from Step 3, plot each pair \((x, f(x))\) on a coordinate plane. This gives the points: \((-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), (3, 5)\).
05

Draw the Graph

After plotting the points, connect them smoothly, drawing a curved line that passes through all the points. The graph will form a parabola opening upwards, with the vertex at the lowest point, \((0, -4)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
Graphing a parabola is a visual way to understand quadratic functions such as \( f(x) = x^2 - 4 \). Parabolas have distinctive U-shaped curves that either open upwards or downwards. In this case, our parabola opens upwards as the coefficient of \( x^2 \) is positive.
\(\)
To graph a parabola, start by plotting several points based on calculated values of \( f(x) \) from different \( x \) values. Points plotted on a graph create a pattern that outlines the shape of the parabola.
\(\)
Here's a list to remember while graphing parabolas:
  • They are symmetrical about the vertical line passing through the vertex.
  • The direction of the opening depends on the sign of the \( x^2 \) coefficient.
  • The vertex provides the parabola's highest or lowest point, depending on its opening.
Connecting these points smoothly will give a precise graph of the quadratic function, emphasizing the curve's symmetry.
Table of Values
Creating a table of values is a step-by-step method to help plot the graph of a function. For the function \( f(x) = x^2 - 4 \), a table of values helps us find specific pairs of \( x \) and \( f(x) \). These pairs form the basis of our plotting process.
\(\)
Here's why a table of values is helpful:
  • It provides organized data, displaying corresponding \( x \) and \( f(x) \) values clearly.
  • The table helps identify patterns in values, showing how \( f(x) \) changes as \( x \) changes.
  • It makes the plotting process on a coordinate system straightforward and systematic.
For instance, choosing \( x \) values from \(-3\) to \(3\) in the original solution helps illustrate how \( f(x) = x^2 - 4 \) manifests as a parabola. Observing how the values form a symmetric outline around the vertex is crucial too.
Vertex of Parabola
The vertex of the parabola is a key point and acts as the "turning point" of the curve. For the function \( f(x) = x^2 - 4 \), the vertex is at \((0, -4)\). This point is the lowest point of the parabola since it opens upwards.
\(\)
The vertex can be thought of as:
  • The point where the parabola changes direction.
  • A balance point of symmetry. Values of \( f(x) \) are mirrored on either side.
  • For \( y = ax^2 + bx + c \), where there's no linear term \( (b = 0) \), the vertex lies on the y-axis.
In our example, the vertex \((0, -4)\) is also the minimum point of the function \( f(x) \). Recognizing the location and significance of the vertex helps in sketching an accurate graph of the quadratic function.

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Most popular questions from this chapter

As a weather balloon is inflated, the thickness T of its rubber skin is related to the radius of the balloon by $$T(r)=\frac{0.5}{r^{2}}$$ where \(T\) and \(r\) are measured in centimeters. Graph the function \(T\) for values of \(r\) between 10 and \(100 .\)

Express the function in the form \(f \circ g \circ h\) $$ Q(x)=(4+\sqrt[3]{x})^{9} $$

Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x^{2}, \quad g(x)=\sqrt{x-3} $$

Sketch graphs of the functions \(f(x)=\|x\|, g(x)=[2 x \|, \text { and } h(x)=\|3 x\| \text { on separate }\) graphs. How are the graphs related? If \(n\) is a positive integer, what does the graph of \(k(x)=\|n x\|\) look like?

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ g(x)=\sqrt{x+3} $$
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