Problem 49 - Algebra and Trigonometry

Question

Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{4} & {\text { if } x<-2} \\ {x^{2}} & {\text { if }-2 \leq x \leq 2} \\ {-x+6} & {\text { if } x>2}\end{array}\right. $$

Step-by-Step Solution

Answer

A horizontal line at \( y = 4 \) when \( x < -2 \), an upward-opening parabola from \( x = -2 \) to \( x = 2 \), and a line with slope -1 for \( x > 2 \).

Step 1: Understand the Function Definition

The function \( f(x) \) is defined piecewise, which means it has different expressions for different intervals of \( x \). We need to evaluate each of these expressions separately over their specified intervals to sketch the graph.

Step 2: Plot the Constant Part

For \( x < -2 \), the function is constant: \( f(x) = 4 \). This means for any value of \( x \) less than -2, \( f(x) \) equals 4. Plot a horizontal line at \( y = 4 \) extending leftward from \( x = -2 \).

Step 3: Plot the Quadratic Part

For \(-2 \leq x \leq 2\), the function is \( f(x) = x^2 \). This is a parabola opening upwards. Calculate some key points: for example, at \( x = -2 \), \( f(x) = (-2)^2 = 4 \); at \( x = 0 \), \( f(x) = 0 \); at \( x = 2 \), \( f(x) = 2^2 = 4 \). Plot these points and connect them with a smooth curve.

Step 4: Plot the Linear Part

For \( x > 2 \), the function is \( f(x) = -x + 6 \). This is a straight line with a slope of -1. Find the point at \( x = 2 \), which is the border of the interval: \( f(2) = -2 + 6 = 4 \). Next, find another point, such as at \( x = 3 \), \( f(3) = -3 + 6 = 3 \). Plot these points and draw the line extending rightwards.

Step 5: Combine the Intervals

Once each interval is plotted, combine them. Ensure that you correctly show open or closed circles at the endpoints: \(-2\) and \( 2 \) both have closed circles because the quadratic function is inclusive of these points. Remember, the sections should blend smoothly if they meet.

Key Concepts

Quadratic Functions

Quadratic functions form one of the most common types of polynomial functions. They take the general form of \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. A characteristic trait of quadratic functions is their U-shaped graph called a parabola. The direction in which the parabola opens depends on the sign of \( a \): if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards.

In our piecewise function, the quadratic part is \( f(x) = x^2 \) for the interval \(-2 \leq x \leq 2\). This parabola opens upwards as the coefficient of \( x^2 \) is positive. By substituting \( x \) with values like \(-2, 0, \text{and } 2\), we discover the corresponding \( f(x) \) values to be 4, 0, and 4, respectively. These points plot the curve's path, creating a smooth upward curve around the x-axis in the designated interval.

Understanding the nature of quadratic functions aids in sketching graphs and predicting their behavior in different domains.

Constant Functions

Constant functions are perhaps the simplest forms of functions where the function's value remains unchanged regardless of the input. They are typically expressed as \( f(x) = c \), where \( c \) is a constant. As such, their graphs are horizontal lines on the coordinate plane. Since every value of \( x \) produces the same output, it's quite straightforward to draw these types of graphs.

In the piecewise function provided, the equation \( f(x) = 4 \) for \( x < -2 \) illustrates a constant function. This indicates that, for any \( x \) less than \(-2\), \( f(x) \) remains consistently 4. On a graph, this is visualized as a horizontal line along \( y = 4 \) beginning from \( x = -2 \) and stretching leftwards.

• Easy to graph: just a straight line at a constant \( y \)-value.
• No slopes or curves as the output never changes.

A solid grasp of constant functions makes it easy to plot and identify this part of any piecewise function.

Linear Functions

Linear functions are fundamental in mathematics and take the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) determines the angle and direction of the line: positive slopes ascend, while negative slopes descend. Linear functions yield a straight line when graphed and perfectly embody the principle of constant rates of change.

For the piecewise function in question, the linear segment is \( f(x) = -x + 6 \) for \( x > 2 \). This function has a slope \( m = -1 \), indicating a downward slant. To graph:

• Evaluate \( f(x) \) at the border, \( x = 2 \), yielding \( f(2) = 4 \).
• Find another point by checking a value like \( x = 3 \); here, \( f(3) = 3 \).

Connecting these points produces a straight line extending to the right. Understanding and identifying linear functions helps students anticipate the steepness and direction the lines take on the graph.