Question

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=1-x^{3}$$

Short answer

The function \(f(x) = 1-x^{3}\) is not symmetric. It has intercepts at x = 1 and y = 1. The function is positive when x < 1 and negative when x > 1. The hand-drawn graph starts in the positive quadrant, crosses the x-axis at x = 1 and continues into the negative quadrant. The function is confirmed with a graphing utility.

Step-by-step solution

Symmetry

To calculate symmetry, replace \(x\) by \(-x\) in the function and simplify. If the simplified function returns to the original function, then the function is even. If it is opposite of the original function, it's odd. For \(f(x)=1-x^{3}\), replacing \(x\) by \(-x\) we get \(f(-x) = 1-(-x)^{3} = 1+x^3\), which is not the same as the original function, hence it is not symmetric about the y-axis.

Intercepts

To find x-intercept, set \(f(x) = 0\) and simplify for \(x\). Hence, \(0 = 1 - x^3 \Rightarrow x^3 = 1, x = 1\). This means the x-intercept is at x = 1. For the y-intercept, set \(x = 0\) in the function: \(f(0) = 1\). Therefore, the y-intercept is at y = 1.

Function's positivity/negativity

After finding the x-intercept, we know that the function equals zero at x = 1. Now to find out where the function is positive or negative, choose a test point to the left and right of x = 1 and compute the function's value. For x > 1 (e.g., x = 2), we have \(f(x) = 1 - (2)^3 = -7\) which is negative. For x < 1 (e.g., x = 0), we have \(f(x) = 1 - (0)^3 = 1\) which is positive. So, the function is positive for x < 1 and negative for x > 1.

Sketch the graph

Based on the analysis, begin by marking the intercepts on the graph. As the function is neither even nor odd, there is no symmetry. The curve starts from the positive section (for x < 1), crosses the x-axis at x = 1 (the intercept), then continues into the negative section (for x > 1).

Check using a graphing utility

Finally confirm your sketch using a graphing utility. Graph the function \(f(x) = 1 - x^{3}\) and see if it matches up with your hand-drawn graph

Key concepts

Graph Symmetry

Understanding the symmetry of graphs is integral to analyzing functions. Symmetry refers to the balance of a graph on either side of an axis or a point. In algebraic terms, a function is symmetric about the y-axis if it's an even function, meaning if you replace every instance of x with -x in the function's equation and the result is the original function, as seen when analyzing f(x)=1-x^3. However, in this case, plugging in -x we get f(-x) = 1+x^3, which isn't equivalent to f(x); therefore, the graph is not symmetric about the y-axis. There's no x-axis or origin symmetry since those would require f(x) = f(-x) and f(x) = -f(-x), respectively, which are not true here. This knowledge is crucial when sketching because it informs whether one side of the graph is a mirrored version of the other.

X-intercept and Y-intercept

Finding where a graph crosses the axes helps to determine its overall shape. An x-intercept is where the graph crosses the x-axis, which is found by setting f(x) = 0 and solving for x. For the function f(x) = 1-x^3, setting f(x) = 0 yields an intercept at x = 1. Similarly, the y-intercept occurs where the graph crosses the y-axis; it is figured out by evaluating f(0), resulting in y = 1 for this function. Marking these points on a graph is one of the first steps to sketch the curve’s path.

Function Positivity and Negativity

Evaluating where a function is positive or negative is crucial for graphing. The sign of f(x) tells us if the graph is above or below the x-axis. After determining the x-intercept of f(x) = 1-x^3 is at x = 1, we can test values around this intercept. By choosing a number greater than 1, let's say 2, we find that f(2) = -7, indicating the graph is below the x-axis for x > 1. Conversely, choosing a number less than 1, like 0, yields f(0) = 1, showing the function is above the x-axis when x < 1. These evaluations communicate the portrait of the function's highs and lows.

Sketching Graphs by Hand

After understanding symmetry, intercepts, and where a function is positive or negative, we can sketch a graph manually. Start by plotting the intercepts, and use the information on positivity and negativity to indicate the parts of the graph above or below the x-axis. For f(x) = 1 - x^3, the graph passes through (1,0) and (0,1) and switches from positive to negative as x increases from 0 to 1. By connecting these points and considering the end behavior (as x approaches infinity, y will go to negative infinity), we can draw a rough depiction of the curve. This visual aid helps students grasp the function's behavior before finalizing the graph with precision.

Using Graphing Utilities

Today's technology offers powerful tools for visualizing algebraic functions. Graphing utilities such as calculators and software applications can confirm our hand-drawn sketches. After analyzing f(x) = 1-x^3 algebraically, we can use a graphing tool to plot the same function. This step acts as a check against our manual sketch. Aligning points of interest, like the intercepts and regions of positivity and negativity, ensures our understanding is correct. These utilities are especially helpful for visual learners and can solidify the link between the algebraic manipulation and graphical interpretation.