Problem 20 - Algebra Form and Function

Question

The form of the expression for the function tells you a point on the graph and the slope of the graph. What are they? Sketch the graph. $$ f(t)=4-2(t+2) $$

Step-by-Step Solution

Answer

#Answer# The slope of the function \(f(t) = 4 - 2(t+2)\) is -2, and a point on the graph is (0,0). The graph is a downward-sloping line passing through the origin with a slope of -2.

Step 1: Simplify the function to slope-intercept form

Transform the given function \(f(t)=4-2(t+2)\) to the slope-intercept form (\(y = mx + b\)), where \(m\) is the slope, and \(b\) is the y-intercept. To do this, simplify the equation: \(f(t)=4-2(t+2)\) \(f(t)=4-2t-4\) \(f(t)=-2t\) Now, the function is in the form \(f(t)=-2t+0\).

Step 2: Determine the slope and a point on the graph

In the simplified form, we can see that the slope \(m=-2\) and the y-intercept \(b=0\). This means that when \(t=0\), \(f(t)=0\). Thus, the given point on the graph is \((0,0)\).

Step 3: Sketch the graph

Using the slope and point found in the previous steps, we can now sketch the graph. Begin at the point \((0,0)\), and use the slope \(-2\) to determine the direction of the line. The slope of \(-2\) indicates that the line should go down one unit and then right one unit. Plot at least two more points on the graph following the slope and then connect the points to create a straight line. The graph will be a downward-sloping line with a slope of \(-2\) and passing through the point \((0,0)\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is fundamental when working with linear functions. The general formula for this form is \(y = mx + b\), where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the function given in the problem, \(f(t) = 4 - 2(t+2)\), our goal is to express it in the slope-intercept form. Simplifying this equation involves expanding the terms in the parentheses and rearranging them, so it becomes easier to identify both the slope and the y-intercept. After simplifying, we have \(f(t) = -2t + 0\). It becomes clear that the slope \(m\) is \(-2\) and the y-intercept \(b\) is \(0\). This simple equation gives us a lot of information about the behavior and position of the line on a graph.

Simplifying Expressions

Simplifying expressions is an essential skill when dealing with linear equations. It involves re-writing expressions in a simpler or more convenient form. Let's demonstrate this with our example, \(f(t) = 4 - 2(t+2)\). To simplify it, begin with distributing the \(-2\) across the terms inside the parentheses:

• \(-2(t+2)\) becomes \(-2t - 4\).

Now, the expression looks like this: \(f(t) = 4 - 2t - 4\). By combining like terms, the 4 and \(-4\), which cancel each other out, you further simplify it to \(f(t) = -2t\). This step helps us see what's effectively happening with the original function and is a crucial part of converting to slope-intercept form. Simplifying expressions helps in better understanding and visualizing mathematical functions, making it easier to graph and solve problems.

Graphing Linear Functions

Graphing linear functions involves plotting a straight line based on the slope and y-intercept given in the slope-intercept form \(y = mx + b\). For the simplified function \(f(t) = -2t\) or \(y = -2x + 0\), the process is straightforward:

• The y-intercept is \(0\), so the line will pass through the origin, or the point \((0, 0)\).
• The slope \(-2\) means that for every 1 unit we move to the right along the x-axis, the line moves down 2 units on the y-axis.

To graph this function, start by plotting the y-intercept at \((0, 0)\). Next, use the slope to identify another point. From \((0, 0)\), move down 2 units and to the right 1 unit, reaching \((1, -2)\). Plot this second point and draw a line through both points. This line represents the graph of the function. Whenever graphed, linear functions produce straight lines, and understanding these simple steps makes it easy to visualize them.