Question

A linear function is given. (a) Find the slope and y-intercept of each function. (b) Use the slope and y-intercept to graph each function. (c) What is the average rate of change of each function? (d) Determine whether each function is increasing, decreasing, or constant. $$ G(x)=-2 $$

Short answer

Slope: 0, y-intercept: -2, Average rate of change: 0, The function is constant.

Step-by-step solution

Identify the linear function

The given function is \(G(x) = -2\). This is a constant function, where the value of the function is always -2 for any input of \(x\).

Find the slope

For a constant function \(G(x) = c\), the slope is 0 because the value does not change with respect to \(x\). Thus, the slope of \(G(x) = -2\) is 0.

Find the y-intercept

The y-intercept is the value of the function when \(x = 0\). Since \(G(x) = -2\), the y-intercept is -2.

Graph the function

To graph \(G(x) = -2\), draw a horizontal line that crosses the y-axis at -2. This represents the constant value of the function across all \(x\).

Calculate the average rate of change

The average rate of change for a linear function \(f(x)\) over the interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\). For \(G(x) = -2\), \(f(b) = -2\) and \(f(a) = -2\), therefore the average rate of change is \(\frac{-2 - (-2)}{b - a} = 0\) for any \(a\) and \(b\).

Determine the nature of the function

Since the slope is 0, the function is constant. Hence, \(G(x)\) is neither increasing nor decreasing; it is a constant function.

Key concepts

slope

The slope is an essential concept when dealing with linear functions. It measures the steepness or incline of a line and is usually represented by the letter 'm'. The slope is calculated by the change in the y-values divided by the change in the x-values between two points on the line. Mathematically, it is expressed as:
ewlineewline\text{Slope} = m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
ewlineewlineFor the function given in the exercise, \(G(x) = -2\), it's a constant function. This means the slope is 0 because the value of the function does not change regardless of \(x\). In other words, the line does not rise or fall but remains flat across the graph. So:
ewlineSlope of \(G(x) = -2\) is \(0\).

y-intercept

The y-intercept of a linear function is the point where the line crosses the y-axis. It is the value of the function when \(x\) is \(0\).
ewlineewlineFor the function \(G(x) = -2\), the y-intercept is found by evaluating the function at \(x = 0\). Since \(G(x) = -2\) is a constant function, it equals \(-2\) for any value of x, including zero. Therefore, the y-intercept is:
ewlineewline\(G(0) = -2\).
ewlineewlineSo, the y-intercept for \(G(x) = -2\) is \(-2\). This means the line crosses the y-axis at \((0, -2)\).

graphing linear functions

Graphing a linear function involves plotting points and drawing a straight line through them. For typical linear functions of the form \(y = mx + b\), start at the y-intercept \((0, b)\) and use the slope \(m\) to find another point on the line.
ewlineewlineFor the function \(G(x) = -2\), which is constant, the graph is simple.
ewlineewline

• Step 1: Identify the y-intercept, which is \(-2\).
• Step 2: Since the slope is \(0\), the line is horizontal.
• Step 3: Draw a horizontal line that crosses the y-axis at \(-2\).

ewlineewlineThe entire line will lie on \(y = -2\), regardless of the values of \(x\).

average rate of change

The average rate of change of a function over an interval \text(\text{[a, b]}\text) is the change in the function's value divided by the change in \(x\). For linear functions, this is equivalent to the slope. It can be calculated using:
ewlineewline\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}.
ewlineewlineIn the case of the constant function \(G(x) = -2\), \(f(b) = -2\) and \(f(a) = -2\) for any values of \(a\) and \(b\). So:
ewlineewline\text(\frac{-2 - (-2)}{b - a} = 0\text).
ewlineewlineTherefore, the average rate of change is \(0\), showing that there is no change in the function's value over any interval.

constant functions

A constant function is a special type of linear function where the function's value does not change regardless of the input. It is represented by \(f(x) = c\), where \(c\) is a constant. For any \(x\), \(f(x)\) remains the same.
ewlineewlineIn the given exercise, \(G(x) = -2\) is a constant function. This means:
ewlineewline

• The slope is \(0\) because the function does not rise or fall.
• The y-intercept is \(-2\), which is the constant value of \(G(x)\).
• The graph is a horizontal line crossing the y-axis at \(-2\).
• The average rate of change is \(0\).

ewlineewlineSince the slope is zero, the function is neither increasing nor decreasing, confirming that \(G(x)\) is a constant function.