Problem 51
Question
Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x^{3}(x-2)$$
Step-by-Step Solution
Verified Answer
Intercepts: (0,0), (2,0); End behavior: left downward, right upward.
1Step 1: Entering the Function into the Calculator
Begin by accessing the graphing feature on your calculator. Enter the polynomial function \( f(x) = x^3(x-2) \). This requires you to input the expression as \( x^3 \times (x-2) \). Double-check that you've entered the function correctly.
2Step 2: Plotting the Graph
After entering the function, generate the graph by selecting the appropriate option on your calculator. Observe the shape of the graph on the display, ensuring all critical points are visible. If not, adjust the window settings to capture the entire graph effectively.
3Step 3: Identifying the Intercepts
To find the x-intercepts, locate where the graph crosses the x-axis. The function \( f(x) = x^3(x-2) \) crosses the x-axis at the points where \( f(x) = 0 \). These are solved by setting \( x^3(x-2) = 0 \), giving roots at \( x = 0 \) and \( x = 2 \). The y-intercept is found when \( x = 0 \), which is easily calculated as \( f(0) = 0^3(0-2) = 0 \). Thus, the y-intercept is at \( (0, 0) \).
4Step 4: Determining the End Behavior
Observe the behavior of the graph as \( x \) approaches positive and negative infinity. For a polynomial with the highest term \( x^4 \), as \( x \rightarrow \infty \), \( f(x) \rightarrow \infty \) and as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \). This means the left end goes downward, and the right end goes upward.
Key Concepts
End behavior of polynomialsIntercepts of graphsUsing graphing calculators
End behavior of polynomials
Understanding the end behavior of polynomial functions is crucial in graphing them correctly. It describes how the function behaves as the input, or \( x \), moves towards positive and negative infinity. For the polynomial function \( f(x) = x^3(x-2) \), we can determine the end behavior by looking at the leading term of the polynomial when expanded. This is \( x^4 \), since \( x^3 \times x = x^4 \).
For polynomials, the degree and the leading coefficient significantly influence the end behavior. When the degree (here, 4) is even, and the leading coefficient is positive, the graph behaves like \( x^2 \), meaning both ends go in the same direction as they approach infinity. As \( x \rightarrow \pm \infty \), the function follows:
Instead, consider the degree of the polynomial in its unexpanded form, where the highest-degree term affects the behavior. Here, it behaves like \( x^3 \), dictating that the left end (\( x \rightarrow -\infty \)) goes downward while the right end (\( x \rightarrow \infty \)) goes upward. It's essential to note the sign and power when deducing end behavior accurately.
For polynomials, the degree and the leading coefficient significantly influence the end behavior. When the degree (here, 4) is even, and the leading coefficient is positive, the graph behaves like \( x^2 \), meaning both ends go in the same direction as they approach infinity. As \( x \rightarrow \pm \infty \), the function follows:
- \( f(x) \rightarrow \infty \) as \( x \rightarrow \infty \)
- \( f(x) \rightarrow \infty \) as \( x \rightarrow -\infty \)
Instead, consider the degree of the polynomial in its unexpanded form, where the highest-degree term affects the behavior. Here, it behaves like \( x^3 \), dictating that the left end (\( x \rightarrow -\infty \)) goes downward while the right end (\( x \rightarrow \infty \)) goes upward. It's essential to note the sign and power when deducing end behavior accurately.
Intercepts of graphs
Intercepts are critical points on a graph where the function crosses the axes. They provide valuable information about the behavior of the function. For the function \( f(x) = x^3(x-2) \), both x- and y-intercepts come into play. Let's delve into how to find them.
The **x-intercepts** are found by setting \( f(x) \) to zero and solving for \( x \). In this function, setting \( x^3(x-2) = 0 \) results in solutions or roots at:
The **y-intercept** occurs where the graph intersects the y-axis, which happens when \( x = 0 \). Substituting in \( x = 0 \) into the function, \( f(0) = 0^3(0-2) = 0 \), gives us a y-intercept at \((0, 0)\). This convenient overlap with the x-intercept highlights a shared point of origin in this case.
The **x-intercepts** are found by setting \( f(x) \) to zero and solving for \( x \). In this function, setting \( x^3(x-2) = 0 \) results in solutions or roots at:
- \( x = 0 \)
- \( x = 2 \)
The **y-intercept** occurs where the graph intersects the y-axis, which happens when \( x = 0 \). Substituting in \( x = 0 \) into the function, \( f(0) = 0^3(0-2) = 0 \), gives us a y-intercept at \((0, 0)\). This convenient overlap with the x-intercept highlights a shared point of origin in this case.
Using graphing calculators
Graphing calculators are invaluable tools for visualizing polynomial functions. They assist in understanding the function's shape, its intercepts, and how it behaves at different values of \( x \). With these devices, you can accurately draw complex functions that might be tough to sketch by hand. Here's a simple guideline for using graphing calculators effectively:
**Entering the Function**
Start by selecting the graphing mode. Carefully enter the polynomial, such as \( f(x) = x^3(x-2) \). Verify that the equation is correct to avoid errors. Precision in input is key to a successful output.
**Viewing the Graph**
Once the function is entered, generate the graph and observe it on the screen. Ensure that important points, such as intercepts and turning points, are visible. You may need to adjust the window settings both horizontally and vertically to capture the entire graph correctly.
**Analyzing the Graph**
Using the graph display, identify both the x- and y-intercepts directly. Look at how the graph moves as \( x \) increases or decreases, capturing the essence of the end behavior. This visualization is incredibly useful for confirming theoretical calculations and gaining an intuitive feel for the problem.
**Entering the Function**
Start by selecting the graphing mode. Carefully enter the polynomial, such as \( f(x) = x^3(x-2) \). Verify that the equation is correct to avoid errors. Precision in input is key to a successful output.
**Viewing the Graph**
Once the function is entered, generate the graph and observe it on the screen. Ensure that important points, such as intercepts and turning points, are visible. You may need to adjust the window settings both horizontally and vertically to capture the entire graph correctly.
**Analyzing the Graph**
Using the graph display, identify both the x- and y-intercepts directly. Look at how the graph moves as \( x \) increases or decreases, capturing the essence of the end behavior. This visualization is incredibly useful for confirming theoretical calculations and gaining an intuitive feel for the problem.
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