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Chapter 1: Problem 22

Set up a table to sketch the graph of each function using the following values: \(x=-3,-2,-1,0,1,2,3\) \(\quad f(x)=x^{2}+1\)

Short Answer

Expert verified
Create and complete the table, then plot the points to sketch the graph.

Step by step solution

01

Setup a Table

Create a table with two columns, one for the values of \(x\) and the other for \(f(x)\). The first column should include the given values: \(-3, -2, -1, 0, 1, 2, 3\).
02

Apply the Function to Each x

Calculate \(f(x)\) for each value of \(x\) using the formula \(f(x) = x^2 + 1\). This involves squaring each \(x\) and then adding 1.
03

Fill the Table

Complete the table with your calculations:- For \(x = -3\), \(f(x) = (-3)^2 + 1 = 9 + 1 = 10\)- For \(x = -2\), \(f(x) = (-2)^2 + 1 = 4 + 1 = 5\)- For \(x = -1\), \(f(x) = (-1)^2 + 1 = 1 + 1 = 2\)- For \(x = 0\), \(f(x) = 0^2 + 1 = 0 + 1 = 1\)- For \(x = 1\), \(f(x) = 1^2 + 1 = 1 + 1 = 2\)- For \(x = 2\), \(f(x) = 2^2 + 1 = 4 + 1 = 5\)- For \(x = 3\), \(f(x) = 3^2 + 1 = 9 + 1 = 10\)
04

Verify Your Calculations

Double-check each calculation to ensure accuracy. Verify that each square calculation was correct and that 1 was added properly.
05

Sketch the Graph

Using the completed table, plot the points \((x, f(x))\) on a graph. For example, for \(x = -3\), you would plot \((-3, 10)\). Connect these points smoothly as they represent the parabola of the function \(f(x) = x^2 + 1\).

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Key Concepts

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

Function Evaluation
Function evaluation involves finding the output or value of a function based on the given input. For our specific quadratic function, which is defined as \(f(x) = x^2 + 1\), we substitute each given \(x\)-value into the function formula. This substitution lets us compute the function's output (\(f(x)\)) for each \(x\).
  • Given input (\(x\)) is squared. For example, if \(x = -3\), we calculate \((-3)^2\).
  • Add 1 to the result of the squared term, providing the final output. Thus, \((-3)^2 + 1 = 10\).
  • This process is repeated for each \(x\)-value provided.
By evaluating this function for each \(x\)-value, we generate points that we can later use for graphing.
Table of Values
Creating a table of values is an essential step in sketching a graph. Your table serves as a visual aid to organize \(x\)-values and their corresponding \(f(x)\) values.Here's how to set up the table:
  • Make two columns: one for \(x\) and the other for \(f(x)\).
  • Input your \(x\)-values. For our exercise, these are \(-3, -2, -1, 0, 1, 2, 3\).
  • Calculate \(f(x)\) for each, using the function \(f(x) = x^2 + 1\).
The table not only facilitates accurate plotting of points on the graph but also checks if your calculations are consistent. Always double-check the values by redoing the function evaluation to ensure accuracy.
Quadratic Function
A quadratic function is a type of polynomial with the general form \(f(x) = ax^2 + bx + c\). In our exercise, the quadratic function is \(f(x) = x^2 + 1\), which can also be expressed as \(f(x) = 1x^2 + 0x + 1\).Some characteristics of quadratic functions include:
  • The squared term \(x^2\) defines its unique parabolic shape.
  • Quadratic functions produce symmetric graphs. This means the parabola looks the same on both sides of its vertex.
  • The vertex of a basic function like \(f(x) = x^2 + 1\) occurs at the point (0,1).
Understanding these properties helps you predict and understand the behavior of the function when graphing.
Graph Sketching
Graph sketching is the final step where you visually represent the function by plotting its values on a coordinate plane.To sketch the graph:
  • Use the table of values as a guide and plot the points like \((-3, 10)\), \((-2, 5)\), \((-1, 2)\), \( (0, 1)\), \( (1, 2)\), \( (2, 5)\), and \( (3, 10)\).
  • Notice the symmetry about the vertex, in this case, \( (0, 1)\).
  • Connect the plotted points smoothly to form a "U" shaped parabola.
The symmetry of the points around the vertex helps in affirming that the graph is accurate. A complete graph provides a visual understanding of how quadratic functions behave, facilitating a deeper analytical perspective.

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Most popular questions from this chapter

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season. The population can be modeled by \(P(t)=82.5-67.5 \cos ((\pi / 6) t],\) where \(t\) is time in months \((t=0 \text { represents January } 1)\) and \(P\) is population (in thousands). During a year, in what intervals is the population less than \(20,000 ?\) During what intervals is the population more than \(140,000 ?\)

[T] Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In \(2012 \quad(t=0),\) total online holiday sales were \(\$ 42.3\) billion, whereas in 2013 they were \(\$ 48.1\) billion. a. Find a linear function \(S\) that estimates the total online holiday sales in the year \(t\) . b. Interpret the slope of the graph of \(S\) c. Use part a. to predict the year when online shopping during Christmas will reach \(\$ 60\) billion.

For the following exercises, sketch the graph of the logarithmic function. Determine the domain, range, and vertical asymptote. $$ f(x)=\ln (x+1) $$

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation \(y=e^{r t},\) where \(y\) is the percentage of radiocarbon still present in the material, \(t\) is the number of years passed, and \(r=-0.0001210\) is the decay rate of radiocarbon. Find the inverse of the carbon-dating equation. What does it mean? If there is 25\(\%\) radiocarbon, how old is the skeleton?

For the following exercises, solve the logarithmic equation exactly, if possible. $$ \log _{4}(x+5)=0 $$
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