Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{3}+2\)

Step-by-Step Solution

Answer

The graph of the function \(y=x^{3}+2\) rises gradually, without any sharp corners or discontinuities. The y-intercept occurs at (0, 2) and the x-intercept occurs at \(-\sqrt[3]{2}, 0\). The function is not symmetrical

Step 1: Sketching Graph

The graph can be manually sketched by plotting points for variable x and finding the corresponding y using the equation \(y=x^{3}+2\). At least five points including negative, positive and zero values of x would generate an accurate graph.

Step 2: Finding Intercepts

To find the y-intercept, substitute x=0 in the given equation, which yields \(y=0^{3}+2=2\), so the y-intercept is at (0,2). To find the x-intercept(s), the equation is set equal to zero, \(x^{3}+2=0\), solving this gives \(x^{3}=-2\), taking cube root of both sides gives \(x=-\sqrt[3]{2}\). So, the x-intercept is at \(-\sqrt[3]{2}, 0\).

Step 3: Testing for Symmetry

If the function is even, it'll be symmetric about the y-axis. If the function is odd, it'll be symmetric about the origin. Here, replace x by -x in the equation, we get \(y=(-x)^{3}+2 = -x^{3}+2\), which is not equal to the original function. So, the function is neither even nor odd, hence it's not symmetric about the y-axis or the origin.

Key Concepts

Intercepts

When sketching the graph of a function, finding intercepts is crucial. Intercepts are points where the graph crosses the axes. There are two types of intercepts: the x-intercept and the y-intercept.

Finding the Y-Intercept:

• Set the value of x to 0 in the equation.
• Calculate the corresponding y This will give the y-intercept, where the graph crosses the y-axis.

For the equation \(y = x^3 + 2\),
If we substitute 0 for x, we get \(y = 0^3 + 2 = 2\). Hence, the y-intercept is (0, 2).

Finding the X-Intercept(s):

• Set the value of y to 0 in the equation.
• Solve for x to find the x-intercepts, where the graph intersects the x-axis.

For the same function, \(x^3 + 2 = 0\).
Solving this gives us \(x = -\sqrt[3]{2}\), so the x-intercept is at \(-\sqrt[3]{2}\, 0\).
Understanding intercepts helps in sketching an accurate graph and gives a clear picture of where the graph cuts the axes.

Symmetry

Symmetry in graphs can tell us a lot about the function's behavior without needing a detailed point-by-point sketch. The two common symmetries we look for are with respect to the y-axis (even functions) or the origin (odd functions).

Testing for Symmetry:

• Even Functions: If a function is even, it is symmetrical about the y-axis. Mathematically, this means that \(f(x) = f(-x)\).
• Odd Functions: If a function is odd, it is symmetrical about the origin. This can be tested by checking if \(f(-x) = -f(x)\).

In the function \(y = x^3 + 2\),
we substitute \(-x\) for \(x\) and find

\(y = (-x)^3 + 2 = -x^3 + 2\).

Since \(-x^3 + 2\) is not equivalent to \(x^3 + 2\) or \(-x^3 - 2\), the function is neither even nor odd.

Checking for symmetry helps quickly understanding the nature of the graph's layout, saving time during plotting and making certain predictions easier.

Cubic Functions

Cubic functions are a type of polynomial function with the general form \(f(x) = ax^3 + bx^2 + cx + d\). These functions can produce a variety of graph shapes, often with a characteristic 'S-curve' due to their highest degree term \(x^3\).

Features of Cubic Functions:

• Degree: Cubic functions are of degree 3, meaning the highest power of \(x\) is 3.
• Intersections: Generally have one real root, but can have up to three real roots, leading to as many x-intercepts.
• End Behavior: On large scales, the graph of a cubic function behaves like \(y = x^3\), typically rising or falling steeply as \(x\) moves away from 0.

For the function \(y = x^3 + 2\),
there is a constant shift upwards by 2, causing the "center" of the cubic function to move but keeping the overall cubic shape.

Understanding cubic functions and their properties helps in predicting the shape and characteristics of their graphs, including bends, twists, and overall direction of the curve.