Problem 52
Question
For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x(x-3)(x+3) $$
Step-by-Step Solution
Verified Answer
X-intercepts: -3, 0, 3; Y-intercept: 0; Rises to the right, falls to the left.
1Step 1: Identify the Polynomial
The given polynomial is \( f(x) = x(x-3)(x+3) \). This is a cubic polynomial because it involves three linear factors and the highest power of \( x \) is 3.
2Step 2: Use a Calculator to Graph the Polynomial
Use a graphing calculator to plot the function \( f(x) = x(x-3)(x+3) \). You will observe how the graph behaves and where it intersects the axes.
3Step 3: Find the X-Intercepts
The x-intercepts occur where the polynomial equals zero: \( x(x-3)(x+3) = 0 \). This gives the solutions \( x = 0 \), \( x = 3 \), and \( x = -3 \).
4Step 4: Find the Y-Intercept
The y-intercept occurs where \( x = 0 \). Plugging into the function gives \( f(0) = 0(0-3)(0+3) = 0 \). Hence, the y-intercept is \( (0, 0) \).
5Step 5: Determine the End Behavior
The end behavior of the polynomial can be determined by looking at the leading term when expanded, which is \( x^3 \). As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \). This behavior is depicted in the graph as it rises to the right and falls to the left.
Key Concepts
Cubic PolynomialsInterceptsEnd BehaviorGraphing Calculators
Cubic Polynomials
Cubic polynomials are mathematical expressions where the highest degree of the variable is three. This means that the greatest power of the variable, often represented as \( x\), is \( x^3\). A general form of a cubic polynomial can be expressed as \( ax^3 + bx^2 + cx + d\), where \( a\), \( b\), \( c\), and \( d\) are constants, and \( a eq 0\).
Cubic polynomials are important because they can model a wide variety of real-world phenomena, from physics to economics. The graph of a cubic polynomial has a characteristic shape that can include up to two bends or "turning points," making it distinct from linear and quadratic polynomials. Because the highest power is three, the graph of a cubic polynomial is non-symmetrical, potentially crossing the x-axis up to three times. In our example, the polynomial \( f(x) = x(x-3)(x+3) \) is a perfectly balanced cubic polynomial where the behavior can be easily interpreted and visually appealing when graphed.
Cubic polynomials are important because they can model a wide variety of real-world phenomena, from physics to economics. The graph of a cubic polynomial has a characteristic shape that can include up to two bends or "turning points," making it distinct from linear and quadratic polynomials. Because the highest power is three, the graph of a cubic polynomial is non-symmetrical, potentially crossing the x-axis up to three times. In our example, the polynomial \( f(x) = x(x-3)(x+3) \) is a perfectly balanced cubic polynomial where the behavior can be easily interpreted and visually appealing when graphed.
Intercepts
Intercepts are key points where a graph crosses the axes in a coordinate system. Understanding intercepts is fundamental when graphing and interpreting the behavior of functions, including polynomials.
The **x-intercepts** are the points where the graph crosses the x-axis, meaning the y-value is zero. For a cubic polynomial like \( f(x) = x(x-3)(x+3) \), we set the function equal to zero to find these points: \( x(x-3)(x+3) = 0\). Solving this equation, we find three x-intercepts: \( x = 0\), \( x = 3\), and \( x = -3\).
The **y-intercept** occurs where the function crosses the y-axis, meaning the x-value is zero. We find it by evaluating the function at \( x = 0\): \( f(0) = 0(0-3)(0+3) = 0\). Hence, the y-intercept is at the origin, \((0, 0)\). By accurately identifying intercepts, we achieve a better understanding of how the graph interacts with the coordinate plane.
The **x-intercepts** are the points where the graph crosses the x-axis, meaning the y-value is zero. For a cubic polynomial like \( f(x) = x(x-3)(x+3) \), we set the function equal to zero to find these points: \( x(x-3)(x+3) = 0\). Solving this equation, we find three x-intercepts: \( x = 0\), \( x = 3\), and \( x = -3\).
The **y-intercept** occurs where the function crosses the y-axis, meaning the x-value is zero. We find it by evaluating the function at \( x = 0\): \( f(0) = 0(0-3)(0+3) = 0\). Hence, the y-intercept is at the origin, \((0, 0)\). By accurately identifying intercepts, we achieve a better understanding of how the graph interacts with the coordinate plane.
End Behavior
End behavior describes how a polynomial function behaves as the variable \( x\) approaches positive or negative infinity. It's an important aspect of polynomial functions that helps anticipate how the graph will stretch towards the far left or right.
For cubic polynomials, the leading term - which contains the highest power of \( x\) - dictates the end behavior. In our example \( f(x) = x(x-3)(x+3) \), when expanded, yields a leading term of \( x^3\). This positive leading coefficient means:
For cubic polynomials, the leading term - which contains the highest power of \( x\) - dictates the end behavior. In our example \( f(x) = x(x-3)(x+3) \), when expanded, yields a leading term of \( x^3\). This positive leading coefficient means:
- As \( x \to \infty\), \( f(x) \to \infty\): the graph rises to the right.
- As \( x \to -\infty\), \( f(x) \to -\infty\): the graph falls to the left.
Graphing Calculators
Graphing calculators are indispensable tools when studying functions and polynomial graphing in particular. They allow users to quickly visualize the behavior of complex equations, like cubic polynomials, by producing detailed graphs.
When you input a function like \( f(x) = x(x-3)(x+3) \) into a graphing calculator, it handles all the calculations and plots the points to sketch the graph. Observing this visual representation helps in identifying features such as intercepts and end behavior without manual plotting.
Features of graphing calculators include:
When you input a function like \( f(x) = x(x-3)(x+3) \) into a graphing calculator, it handles all the calculations and plots the points to sketch the graph. Observing this visual representation helps in identifying features such as intercepts and end behavior without manual plotting.
Features of graphing calculators include:
- Ability to zoom in and out for a closer look at particular points.
- Options to find specific values or analyze function behavior.
- Quick adjustments to test different factors and transformations.
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