Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ y=(x-1)(x-2)(x-3) $$

Step-by-Step Solution

Answer

The graph is a cubic curve with x-intercepts at \((1,0)\), \((2,0)\), and \((3,0)\), and a y-intercept at \((0,-6)\).

Step 1: Identify the Type of Function

The given equation is in the form of a factored polynomial. It is a cubic function because it has three linear factors: \((x-1)\), \((x-2)\), and \((x-3)\). This means the graph will have a general cubic shape.

Step 2: Determine the X-Intercepts

To find the \(x\)-intercepts, set \(y = 0\) and solve for \(x\). Each linear factor gives an \(x\)-intercept at its root:- \(x-1 = 0 \Rightarrow x = 1\)- \(x-2 = 0 \Rightarrow x = 2\)- \(x-3 = 0 \Rightarrow x = 3\)Thus, the \(x\)-intercepts are at \((1, 0)\), \((2, 0)\), \((3, 0)\).

Step 3: Find the Y-Intercept

To find the \(y\)-intercept, set \(x = 0\):\[ y = (0-1)(0-2)(0-3) = (-1)(-2)(-3) = -6 \]Therefore, the \(y\)-intercept is at \((0, -6)\).

Step 4: Check for Symmetry

Cubic functions like this typically do not have symmetry with respect to the \(y\)-axis or the origin. Verify this by noting that replacing \(x\) with \(-x\) does not yield either even or odd function characteristics for the polynomial.

Step 5: Plot Key Points and Sketch the Graph

Using the intercepts and the general shape of a cubic function:- Plot the \(x\)-intercepts \((1, 0)\), \((2, 0)\), and \((3, 0)\).- Plot the \(y\)-intercept at \((0, -6)\).Begin drawing the cubic curve passing through these points. Note that it will start from negative infinity on the left, turn upward, pass through the intercepts, and finish going to positive infinity on the right.

Key Concepts

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three. The general form of a cubic function is expressed as \(y = ax^3 + bx^2 + cx + d\). In these functions, the output, \(y\), is determined by the input, \(x\), raised to the power of three.

The equation \(y=(x-1)(x-2)(x-3)\) is an example of a cubic function in factored form. Here, it's expressed as a product of linear factors which simplifies identifying roots or x-intercepts. This representation shows the points where the graph will intersect the x-axis.

• A cubic function can have one real root and two complex roots, or three real roots.
• It often has a distinctive S-shaped curve when plotted, which begins at an infinite value and ends at another infinite value on the opposite side.

Understanding these functions is crucial for graphing, as the degree of the function dictates the potential number of turning points and overall behavior.

X-Intercepts

X-intercepts are points where a graph crosses the x-axis, meaning the y-value at these points is zero. To find the x-intercepts of a function, you set the equation equal to zero and solve for \(x\).

For the function \(y=(x-1)(x-2)(x-3)\), setting \(y = 0\) reveals the roots of each factor:

• \(x-1=0\) gives us \(x=1\)
• \(x-2=0\) gives us \(x=2\)
• \(x-3=0\) gives us \(x=3\)

These calculations show that the x-intercepts are at the coordinates \((1, 0)\), \((2, 0)\), and \((3, 0)\). Identifying x-intercepts is important because they indicate where a graph changes direction.

Y-Intercepts

The y-intercept of a function is where it crosses the y-axis, meaning the x-value is zero. To find the y-intercept, substitute \(x = 0\) into the equation and solve for \(y\).

In the equation \(y=(x-1)(x-2)(x-3)\), when you set \(x = 0\), it results in:\[y = (0-1)(0-2)(0-3) = (-1)(-2)(-3) = -6\]This means the y-intercept is at the point \((0, -6)\). Recognizing the y-intercept helps in constructing and understanding the overall trajectory of the graph.

Function Symmetry

Understanding symmetry in functions helps simplify graph analysis and prediction. Symmetry indicates balance and regularity in a graph's shape.

For cubic functions like \(y=(x-1)(x-2)(x-3)\), symmetry isn't always present.

• To check symmetry, replace \(x\) with \(-x\) in the function and observe any changes.
• If replacing \(x\) with \(-x\) yields the same equation or its opposite, the function would be symmetric about the y-axis or the origin, respectively.

For this function, substituting \(x\) with \(-x\) does not return the original or opposite function, suggesting no symmetry. Recognizing the lack or presence of symmetry aids in sketching and verifying polynomial graphs.