Problem 7 - Algebra and Trigonometry

Question

\(5-8\) Sketch the graph of each function by transforming the graph of an appropriate function of the form \(y=x^{n}\) from Figure 2 . Indicate all \(x\) - and \(y\) -intercepts on each graph. $$ \begin{array}{ll}{\text { (a) } P(x)=x^{3}-8} & {\text { (b) } Q(x)=-x^{3}+27} \\\ {\text { (c) } R(x)=-(x+2)^{3}} & {\text { (d) } S(x)=\frac{1}{2}(x-1)^{3}+4}\end{array} $$

Step-by-Step Solution

Answer

Sketch transformations: P is shifted down 8, Q is reflected and up 27, R is reflected and left 2, S is stretched, right 1, and up 4. Intercepts adjusted accordingly.

Step 1: Identify the Base Function

For each function, identify the base function from the form \(y = x^n\).- For \(P(x) = x^3 - 8\), the base function is \(y = x^3\).- For \(Q(x) = -x^3 + 27\), the base is \(y = x^3\).- For \(R(x) = -(x+2)^3\), the base is \(y = x^3\).- For \(S(x) = \frac{1}{2}(x-1)^3 + 4\), the base is \(y = x^3\).

Step 2: Determine Transformations

Determine how each graph is transformed from \(y = x^3\). - For \(P(x) = x^3 - 8\), the graph is shifted down by 8 units.- For \(Q(x) = -x^3 + 27\), the graph is reflected over the x-axis and then shifted up by 27 units.- For \(R(x) = -(x + 2)^3\), the graph is reflected over the x-axis and shifted left by 2 units.- For \(S(x) = \frac{1}{2}(x-1)^3 + 4\), the graph is stretched vertically by a factor of \(\frac{1}{2}\), shifted right by 1 unit, and up by 4 units.

Step 3: Sketch the Graphs

Sketch the graph for each function using the transformations identified:- For \(P(x)\), start with \(y = x^3\) and move it down 8 units.- For \(Q(x)\), reflect \(y = x^3\) across the x-axis and move it up 27 units.- For \(R(x)\), reflect \(y = x^3\) across the x-axis and shift 2 units left.- For \(S(x)\), stretch \(y = x^3\) vertically by \(\frac{1}{2}\), then shift right 1 unit and up 4 units.

Step 4: Find Intercepts

Calculate the x- and y-intercepts for each graph:- For \(P(x)\), set \(P(x)=0\) to find x-intercepts. Solve: \(x^3 - 8 = 0 \Rightarrow x=2\). The y-intercept happens when \(x=0\). Thus, \(P(0)=-8\).- For \(Q(x)\), set \(Q(x)=0\). Solve: \(-x^3 + 27 = 0 \Rightarrow x=3\). The y-intercept is \(Q(0)=27\).- For \(R(x)\), there is no real x-intercept since \(-(x+2)^3=0\) is not possible. The y-intercept is \(R(0)=-(2)^3 = -8\).- For \(S(x)\), set \(S(x)=0\). Solve: \(\frac{1}{2}(x-1)^3 + 4 = 0\). This results in \((x-1)^3 = -8 \Rightarrow x = -1\). The y-intercept is \(S(0)=\frac{1}{2}(-1)^3+4 = \frac{-1}{2}+4 = \frac{7}{2}\).

Step 5: Confirm and Finalize Sketch

Ensure that the transformations and intercept calculations align with the initial equation and sketch. Adjust any incorrect elements in the graph.

Key Concepts

cubic functions

Cubic functions are a type of polynomial function that have the general form of \(f(x) = ax^3 + bx^2 + cx + d\). The defining characteristic of a cubic function is the presence of the \(x^3\) term, which gives the curve its distinct S shape.

Understanding cubic functions is essential when examining the behavior of these functions on a graph. The \(x^3\) term is what distinguishes cubic functions from other polynomial functions, like quadratic \(x^2\) or linear \(x\). This term ensures that as \(x\) approaches infinity, the value of \(f(x)\) also grows infinitely. Conversely, as \(x\) approaches negative infinity, \(f(x)\) becomes infinitely negative. This opposite end behavior is unique to cubic functions and an essential feature when sketching graphs.

The shape of the graph is influenced by the specific coefficients \(a, b, c,\) and \(d\), which control the stretching, translation, and reflection of the basic cubic curve \(y = x^3\). Transformations applied to a cubic function can shift the graph up, down, left, or right, reflect it over the axes, or compress and stretch it vertically or horizontally. These transformations are critical when analyzing and graphing cubic functions.

graph sketching

Graph sketching revolves around the process of outlining a graph by using key features and transformations of a given function. It's an important skill that allows you to visualize how a function behaves without plotting every point.

To sketch a graph of a cubic function, start by identifying the base graph, typically \(y = x^3\). Then, apply transformations based on the changes in the function's equation.

• Shifts: Move the graph up, down, left, or right based on the constants added or subtracted in the function.
• Reflections: If the cubic term is negative, like in \(-x^3\), the graph will reflect over the x-axis.
• Stretches/Compressions: Multiplying the function by a constant \(a\) stretches (\(a>1\)) or compresses (\(0

These transformations are akin to moving or reshaping the graph to fit the function's equation better. Practice these steps, and soon you'll master the art of quick and accurate graph sketching.

intercepts

Intercepts are fundamental components in graphing functions as they denote the points where the graph intersects the axes. There are two primary types of intercepts:

• x-intercepts: These occur where the graph crosses the x-axis (i.e., \(y = 0\)). To find these, set the function equal to zero and solve for \(x\). For example, for \(P(x)=x^3 - 8\), solve \(x^3 - 8 = 0\) to find \(x = 2\).
• y-intercepts: These occur where the graph crosses the y-axis (i.e., \(x = 0\)). To find the y-intercept, calculate \(f(0)\). For instance, for the function \(Q(x) = -x^3 + 27\), \(Q(0) = 27\), giving a y-intercept of 27.

Calculating intercepts provides anchor points to correctly position the sketch of the graph. They help to verify the accuracy of transposed graphs and ensure the complete function's visual model aligns with its analytic form.