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Chapter 1: Problem 14

Graph each equation . Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$y=x^{2}+2$$

Short Answer

Expert verified
The graph of the equation \(y=x^2+2\) is a parabola and the points plotted include \((-3, 11), (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)\)

Step by step solution

01

Understand the equation

We are dealing with a quadratic equation \(y = x^2 + 2\). For every value of \(x\), we can find a corresponding value of \(y\), forming a coordinate \((x, y)\) on the graph. We are given seven distinct values of \(x\) to determine the corresponding \(y\)-values.
02

Substitute the x-values into the equation

We substitute each of the seven values of \(x\) into the equation to find their corresponding \(y\)-values. Using the equation \(y = x^2 + 2\), we find y for each given x. For instance, if \(x = -3\), then \(y = (-3)^2 + 2 = 11\). We continue this for all the given x-values. The variables and their corresponding values would be \((-3, 11), (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)\).
03

Plotting points

Now we have coordinates that can be plotted on a graph. Each pair \((x, y)\) corresponds to a point on the graph. We will locate these points on a graph where the horizontal and vertical lines are marked with numbers. The numbers on the lines are the x-values and y-values.
04

Draw the curve

Once the points are plotted, we draw a smooth curve connecting them. The graph should be a parabola because it represents a quadratic function.

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Key Concepts

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

Graphing Quadratic Equations
Graphing is a way of visually representing mathematical equations. In this case, we have a quadratic equation, which is of the form \(y = x^2 + 2\). When we graph this equation, we want to see how every change in \(x\) affects \(y\). This helps us understand the relationship between \(x\) and \(y\).

Here鈥檚 a simple step-by-step for graphing this equation:
  • First, choose the values of \(x\). In our example, we are using \(x = -3, -2, -1, 0, 1, 2, 3\).
  • Next, substitute each \(x\)-value into the equation \(y = x^2 + 2\) to find the corresponding \(y\)-values.
  • With these pairs, you can mark each coordinate on a graph.
Graphing helps make abstract equations more concrete. It allows us to see patterns and behaviors of equations visually.
Understanding Parabolas
A parabola is a specific type of curve you get from graphing a quadratic equation. In this case, the graph of \(y = x^2 + 2\) forms a special U-shaped curve. This shape is due to the quadratic term \(x^2\).

Every parabola has unique properties:
  • The vertex is the lowest or highest point on the graph. Here, our vertex is at \(x=0, y=2\) because the parabola opens upwards.
  • The axis of symmetry is the vertical line that divides the parabola into two mirror-image halves. In this equation, the line is \(x=0\).
  • Every point on one side of the axis is matched by a point directly opposite on the other side. This symmetry makes parabolas easy to plot.
Understanding the parabola in a quadratic equation helps predict its shape and sketch it accurately.
Coordinate Pairs and Plotting
Coordinate pairs, like \((x, y)\), are essential for plotting any graph. In our example, after substituting \(x\)-values into the equation \(y = x^2 + 2\), you get your \(y\)-values. These results become coordinate pairs.

Here's how you use them:
  • Write down each calculated \((x, y)\) pair. For example, \((-3, 11), (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6), (3, 11)\).
  • Plot each pair on a graph with a horizontal (x-axis) and a vertical line (y-axis).
  • Each point represents a position on the graph that satisfies the equation.
Coordinate pairs are the backbone of graphing. They help us transform algebraic solutions into visual representations, making it easier to understand the behavior of functions.

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Most popular questions from this chapter

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f\left(\frac{1}{2} x\right),\) and \(h(x)=f\left(\frac{1}{4} x\right)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g,\) and \(h,\) with cmphasis on different values of \(x\) for points on all three graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(0

In Exercises \(83-85,\) use a graphing utility to graph each circle whose equation is given. Use a square setting for the viewing window. $$x^{2}+10 x+y^{2}-4 y-20=0$$

Describe a procedure for finding \((f \circ g)(x) .\) What is the name of this function?

What is the graph of a function?

Explain how to identify the domain and range of a function from its graph.
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