Problem 72 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. . \(y=\sqrt[3]{x+1}\)

Step-by-Step Solution

Answer

The graph of the equation \(y = \sqrt[3]{x+1}\) will have intercepts at (0, 1) and (-1, 0) and contain no symmetry.

Step 1: Identify and Plot the Intercepts

The y-intercept is obtained by setting \(x = 0\) in the equation, which yields \(y = \sqrt[3]{0 + 1} = 1\), and the x-intercept is obtained by setting \(y = 0\), which gives \(x = -1\). This gives points (0, 1) and (-1, 0) which can be plotted on the graph.

Step 2: Sketch the Graph

Begin by sketching the points identified as intercepts. To obtain more points for your graph, choose a few arbitrary values for x, calculate the corresponding y, and plot these on the graph. For this equation, the graph will start from the x-intercept and increase gradually as x increases.

Step 3: Test for Symmetry

A function is even if f(x) = f(-x), and odd if f(-x) = -f(x). Substituting -x into the equation gives \(\sqrt[3]{-x+1}\) which is neither \(\sqrt[3]{x+1}\) (even function) nor \(-\sqrt[3]{x+1}\) (odd function). Therefore, the graph has no symmetry.

Key Concepts

Intercepts of a Function

Understanding the intercepts of a function is essential in graphing as it reveals where the graph intersects the axes. To find the y-intercept of a cube root function like y = \(\sqrt[3]{x+1}\), set the x value to zero and solve for y. This gives us the point (0, 1), where the graph touches the y-axis. To find the x-intercept, set the y value to zero and solve for x. For this function, the x-intercept is at (-1, 0) which is where the graph crosses the x-axis.

Remember, the intercepts provide the starting framework for your graph and are crucial for sketching an accurate representation of the function. By plotting these intercepts as your initial points, you can start to visualize the function's behavior easily.

Testing for Symmetry

Symmetry in functions creates a mirror effect across an axis or central point. To determine if the function y = \(\sqrt[3]{x+1}\) is symmetric, test for even or odd symmetry. Here, we observe that when we replace x with -x, we get \(\sqrt[3]{-x+1}\), which is neither \(\sqrt[3]{x+1}\) nor its negative. Therefore, this function has no symmetry. This means when graphing the function, there won't be any reflective properties to rely on, which sets cube root functions apart from their square root counterparts. Knowing the symmetry properties of a function is helpful as it allows for predicting the shape and orientation of a graph before plotting additional points.

Sketching Graphs

To properly sketch cube root functions, start by plotting the intercepts, as mentioned earlier. Then, carefully choose a selection of x values, and compute the respective y values to find additional points. When sketching y = \(\sqrt[3]{x+1}\), begin at the x-intercept (-1, 0), and from there, draw the curve that gently increases as x increases and passes through the y-intercept (0, 1).

The nature of the cube root function means the graph remains smooth and continuous, without any breaks or sharp turns. Unlike parabolas or other polynomial functions, the cube root extends in both directions without any bounds. Hence, the challenge in sketching is to ensure the curve is neither too steep nor too flat but reflects the steady increase or decrease of a cube root function.