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Magazine Chapter 2: Problem 71
Graph each function. $$y=\frac{1}{2} x^{3}-4$$
Short Answer
Expert verified
Plot the points (0, -4), (2, 0), and (1, -3.5) to graph \(y = \frac{1}{2} x^3 - 4\).
Step by step solution
01
- Identify the Function Type
The given function is a cubic function because the highest power of the variable x is 3. Cubic functions have the general form: \(y = ax^3 + bx^2 + cx + d\).
02
- Determine Key Features
For the function \(y = \frac{1}{2} x^3 - 4\), identify key points such as the x-intercepts, y-intercept, and the behavior of the graph. Note that there is no bx^2 or cx term in this function.
03
- Find the Y-Intercept
To find the y-intercept, set \(x = 0\):\(y = \frac{1}{2} (0)^3 - 4 = -4\).So the y-intercept is at (0, -4).
04
- Find the X-Intercepts
Set \(y = 0\) and solve for x:\(0 = \frac{1}{2} x^3 - 4\)\(\frac{1}{2} x^3 = 4\)\(x^3 = 8\)\(x = \root {3}{8}\)\(x = 2\).So the x-intercept is at (2, 0).
05
- Plot Additional Points
Choose additional values for x and solve for y to plot more points. For example, for \(x = 1\): \(y = \frac{1}{2}(1)^3 - 4 = \frac{1}{2} - 4 = -\frac{7}{2}\).Thus, (1, -3.5) is another point on the graph.
06
- Plot the Graph
With the points (0, -4), (2, 0), and (1, -3.5), plot the function on a coordinate plane. Sketch the curve that passes through these points, noting that as x becomes very large or very small, the cubic term \(\frac{1}{2} x^3\) will dominate the behavior of the graph.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cubic Functions
A cubic function is a type of polynomial where the highest exponent of the variable (typically 'x') is 3. The general form of a cubic function is given by:
\(y = ax^3 + bx^2 + cx + d\).
Cubic functions can have both local maxima and minima, which means they can change direction up to two times. They also generally have a unique shape, characterized by one inflection point鈥攖he point where the graph changes curvature.
In our specific function
, \(y = \frac{1}{2} x^3 - 4\), there are no quadratic ( \(bx^2\)) or linear ( \(cx\)) terms. This simplifies our function, making it easier to identify key features.
\(y = ax^3 + bx^2 + cx + d\).
Cubic functions can have both local maxima and minima, which means they can change direction up to two times. They also generally have a unique shape, characterized by one inflection point鈥攖he point where the graph changes curvature.
In our specific function
, \(y = \frac{1}{2} x^3 - 4\), there are no quadratic ( \(bx^2\)) or linear ( \(cx\)) terms. This simplifies our function, making it easier to identify key features.
X-Intercepts
X-intercepts are the points where the graph intersects the x-axis. To find these points, set \(y = 0\) in the cubic function and solve for \(x\). For example: 0 = \( \frac{1}{2} x^3 - 4\). Simplify this equation step-by-step:
- Add 4 to both sides: \( \frac{1}{2} x^3 = 4\)
- Multiply both sides by 2: \(x^3 = 8\)
- Take the cube root of both sides: \(x = 2\)
Y-Intercepts
The y-intercept is the point where the graph crosses the y-axis. To find this point, set \(x = 0\) in the cubic function and solve for \(y\). For our cubic function, \(y = \frac{1}{2} (0)^3 - 4\):
- \(0^3\) is 0, so it simplifies to: \(y = -4\)
Plotting Points
When graphing a cubic function, plotting additional points helps in accurately sketching the curve. Here's how to do it:
- Choose a value for \(x\)
- Substitute this value into the function to solve for \(y\)
- Substitute 1 in:\(y = \frac{1}{2}(1)^3 - 4\)
- \( (1)^3 \) is 1; so, \(\frac{1}{2} \times 1 = \frac{1}{2}\)
- \( \frac{1}{2} - 4\) simplifies to \(-\frac{7}{2} \) or -3.5
