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Magazine Chapter 2: Problem 15
Sketch the graph of the function by first making a table of values. $$g(x)=x^{3}-8$$
Short Answer
Expert verified
Plot key points and draw the curve based on the table values of \(g(x)\).
Step by step solution
01
Determine Key Values of x
To graph the function, we need to determine key points that will help us understand its shape. Choose values of x that are both negative and positive, as well as zero. Typical choices might be \(x = -2, -1, 0, 1, 2, 3\).
02
Calculate g(x) for Each x
Using the formula \(g(x) = x^3 - 8\), compute \(g(x)\) for each selected value of \(x\). For example:- When \(x = -2\), \(g(x) = (-2)^3 - 8 = -8 - 8 = -16\).- When \(x = -1\), \(g(x) = (-1)^3 - 8 = -1 - 8 = -9\).- When \(x = 0\), \(g(x) = (0)^3 - 8 = 0 - 8 = -8\).- When \(x = 1\), \(g(x) = (1)^3 - 8 = 1 - 8 = -7\).- When \(x = 2\), \(g(x) = (2)^3 - 8 = 8 - 8 = 0\).- When \(x = 3\), \(g(x) = (3)^3 - 8 = 27 - 8 = 19\).
03
Construct the Table of Values
Based on the calculations from Step 2, create a table of values showing \(x\) and the corresponding \(g(x)\):\[\begin{array}{|c|c|}\hlinex & g(x) \\hline-2 & -16 \-1 & -9 \0 & -8 \1 & -7 \2 & 0 \3 & 19 \\hline\end{array}\]
04
Plot the Points on a Graph
Use the table of values to plot corresponding points on a coordinate plane. For example, plot the points \((-2, -16)\), \((-1, -9)\), \((0, -8)\), \((1, -7)\), \((2, 0)\), and \((3, 19)\) on the graph.
05
Draw the Curve
After plotting the points, draw a smooth curve through them to represent the function \(g(x) = x^3 - 8\). Ensure the curve accurately reflects the shape indicated by your plotted points, noticing particularly the trend and direction of the line as \(x\) increases or decreases.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Table of Values
Creating a table of values is an essential first step when sketching the graph of a function. This table acts like a blueprint, giving us a clearer idea of the function's behavior at specific points. You start by selecting key values for the variable, usually denoted by \(x\). For cubic functions like \(g(x) = x^3 - 8\), it's vital to choose both negative and positive values, as well as zero, to fully capture the symmetry and range of the function.
Here's how this works in practice:
Here's how this works in practice:
- Select specific \(x\) values. In our example, these are \(-2, -1, 0, 1, 2, 3\).
- Calculate \(g(x)\) for each of these \(x\) values using the given function formula. Substitute each \(x\) into the function to find corresponding \(g(x)\) values.
Plotting Points on a Graph
With your table of values complete, the next step is to bring these numbers to life by plotting the points on a graph. This step helps you start visualizing how the function behaves and its general shape.
To plot the points:
To plot the points:
- Use a coordinate plane, which consists of two axes; the horizontal \(x\)-axis and the vertical \(y\)-axis.
- For each ordered pair \((x, g(x))\) from your table, locate the \(x\) coordinate on the \(x\)-axis and the \(g(x)\) coordinate on the \(y\)-axis.
- Place a dot where these two values meet on the graph.
Function Transformation
Understanding function transformation is crucial in recognizing how changes in equations affect their graphs. In \(g(x) = x^3 - 8\), the function \(x^3\) is the base cubic function, while \(-8\) is a transformation that shifts the entire graph downward.
Function transformation can occur in various forms:
Function transformation can occur in various forms:
- Vertical Shifts: Adding or subtracting a number moves the graph up or down.
- Horizontal Shifts: Changes inside the function (before applying operations) move the graph left or right.
- Reflections: Negative signs can flip the graph over an axis.
- Stretch/Compress: Multiplying by a constant changes the graph's steepness or width.
- The "-8" indicates a vertical shift of 8 units downward.
Smooth Curve Drawing
Once you've plotted all the key points on your graph, the next task is to connect these dots with a smooth curve. This curve represents the function's continuous behavior across the selected range of \(x\) values.
Here are some steps to drawing a smooth curve:
Here are some steps to drawing a smooth curve:
- Begin at one end of your plotted points and move steadily to the other end, following the direction implied by your points.
- Aim for a seamless connection, avoiding any sharp angles - the curve should flow smoothly through each point.
- Observe the shape - for a cubic function like \(g(x) = x^3 - 8\), expect the curve to have a gentle s-like shape with a point of inflection near \(x=0\).
