Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Graph the functions by starting with the graph of a familiar function and applying appropriate shifts, flips, and stretches. Label all \(x\) - and \(y\) -intercepts and the coordinates of any vertices and corners. Use exact values, not numerical approximations. (a) \(y=-x+\pi\) (b) \(y=-(x+\pi)\)

Step-by-Step Solution

Answer

For \(y=-x+π\), the x-intercept and y-intercept are both \(π\). For \(y=-(x+π)\), the x-intercept and y-intercept are both -\(π\).

Step 1: Draw the Graph of \(y=-x+π\)

Start by recognizing that this is a linear function, and assume the form of y=mx+c where m is the gradient and c is the y-intercept. Here, the gradient m is -1 and the y-intercept c is \(π\). To plot, begin at the y-intercept, which essentially is (0, \(π\)). Since the gradient is -1, it means for each step to the right on the x-axis, take one step down on the y-axis. Remember to indicate the y-intercept. For x-intercept, set y = 0 in the equation which will yield \(x = π\). For y-intercept, set x = 0 which gives \(y=π\). Hence, x-intercept is \(π\) and y-intercept is \(π\).

Step 2: Draw the Graph of \(y=-(x+π)\)

This is another linear function. Here, it's important to note that the equation can be rewritten as \(y=-x-π\). To plot this function, observe that the gradient is -1 and the y-intercept is -\(π\). Starting from the y-intercept (0, -\(π\)) is recommended as it's the point where the graph will intersect the y-axis. Then, for every positive step to the right on the x-axis, take a negative step down on the y-axis because of the negative gradient. For x-intercept, set y = 0 in the equation, which yields \(x = -π\). For y-intercept, set x = 0 which gives \(y = -π\). So, the x-intercept equals -\(π\) and the y-intercept equals -\(π\).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions you can encounter in mathematics. It's represented by the equation \( y = mx + c \), where \( m \) is the slope or gradient, and \( c \) is the y-intercept. These functions create straight lines on a graph. The slope \( m \) tells us how steep the line is.

• When \( m > 0 \), the line rises as you move from left to right.
• When \( m < 0 \), the line falls as you move from left to right.

The y-intercept \( c \) is where the line crosses the y-axis. For example, in the equation \( y = -x + \pi \), the slope is \(-1\), indicating that the line slopes downward, and the y-intercept is \( \pi \), which is where the line meets the y-axis.

Graph Transformations

Graphs of functions can be transformed in several ways, including shifting, reflecting, stretching, or compressing. These modifications alter the appearance but not the fundamental shape.

• Shifts: Moving a graph up, down, left, or right without changing its shape. For example, \( y = f(x) + c \) shifts the graph up by \( c \) units.
• Reflections: Flipping the graph over an axis. The graph of \( y = -f(x) \) is a reflection of \( y = f(x) \) over the x-axis.
• Stretching/Compressing: Changing the graph's width or height. A vertical stretch is when \( y = a \cdot f(x) \) and \( a > 1 \).

In the exercise, the transformation from \( y = -x + \pi \) to \( y = -(x + \pi) \) involves both shifting and reflecting.

Intercepts in Graphing

Intercepts are crucial in understanding where a graph crosses the axes.

• x-intercept: The point where the graph crosses the x-axis, found by setting \( y = 0 \) and solving for \( x \).
• y-intercept: The point where the graph crosses the y-axis, found by setting \( x = 0 \) and solving for \( y \).

In the exercise, for \( y = -x + \pi \), both the x-intercept and y-intercept occur at \( \pi \). For \( y = -(x + \pi) \), they are both found at \(-\pi\). Recognizing intercepts helps when sketching graphs.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses algebra to describe geometry through coordinates. The coordinates \((x, y)\) represent points on the plane.

• Plotting Points: You determine where to place a point using its coordinates. For example, \((0, \pi)\) is a position on the y-axis.
• Lines and Slopes: Lines can be defined by points and slopes. The equation \( y = mx + c \) describes these lines.

By understanding coordinate geometry, you can easily graph linear functions. Each point plotted can tell a story about the function you are dealing with. In our case, knowing how to plot both functions leads to accurate and insightful graphs.