Problem 19 - 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. $$ f(x)=3(x-1)+5 $$

Step-by-Step Solution

Answer

Answer: The slope of the function is 3, and the y-intercept is 2. A point on the graph is (0, 2).

Step 1: Identify the slope and y-intercept

The given function is f(x) = 3(x - 1) + 5. Comparing it with the slope-intercept form y = mx + b, we can identify that the slope (m) is 3, and the y-intercept (b) can be found by simplifying the expression.

Step 2: Simplify the expression

Now, we will simplify f(x) to find the y-intercept (b). f(x) = 3(x - 1) + 5 f(x) = 3x - 3 + 5 f(x) = 3x + 2 Now, we can see that y-intercept (b) is 2.

Step 3: Find a point on the graph

We know that the y-intercept is a point on the graph where x = 0. So, to find the coordinates of the point, we will plug x = 0 into the simplified function: f(0) = 3(0) + 2 f(0) = 0 + 2 f(0) = 2 The point on the graph is (0, 2).

Step 4: Sketch the graph

To sketch the graph of the function f(x) = 3x + 2, start by plotting the point (0, 2) on the graph. This is the y-intercept, where the graph intersects the y-axis. Next, use the slope to plot more points on the graph. Since the slope is 3, it means that for each unit increase in x, the value of f(x) increases by 3 units. From the point (0, 2), move 1 unit to the right and 3 units up to find another point on the graph, which is (1, 5). Plot this point and repeat the process to find additional points on the graph. Finally, use a ruler or freehand to draw a straight line that passes through the plotted points. This line represents the graph of the function f(x) = 3x + 2.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a convenient way to express the equation of a line. It comes in handy because it clearly shows both the slope and the y-intercept of the line. The general structure of slope-intercept form is given by the equation \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept.
The slope \( m \) indicates how steep the line is, showing how much \( y \) changes for a unit change in \( x \). Meanwhile, the y-intercept \( b \) is the point where the line crosses the y-axis. It's where \( x = 0 \).
In our exercise, the function \( f(x)=3(x-1)+5 \) was simplified to \( f(x)=3x+2 \). Comparing with \( y=mx+b \), we determine that the slope \( m \) is 3 and the y-intercept \( b \) is 2.

Graph Sketching

Graph sketching is an essential skill that allows us to visualize mathematical equations and their solutions. To sketch the graph of a linear function like \( f(x) = 3x + 2 \), follow a straightforward method.

First, identify and plot the y-intercept. For our equation, that's the point (0, 2), which you mark on the graph where the line will cross the y-axis.
Next, use the slope. Remember, the slope is 3, meaning for each increase of 1 in \( x \), \( y \) increases by 3. Start at the y-intercept (0, 2), then move 1 unit to the right (x-direction) and 3 units up (y-direction). This gives you another point: (1, 5).

• Plot the point (0, 2).
• From (0, 2), go 1 unit right and 3 units up to plot (1, 5).

Draw a line through these points, extending it across the graph. This is your linear function visualized with points neatly laid out.

Y-Intercept

The y-intercept of a function is crucial when understanding or sketching its graph. It essentially acts as the starting point since it's the value of \( y \) when \( x = 0 \). This feature of linear functions makes it easy to quickly identify the initial point of the graph on the y-axis.
In our simplified function \( f(x) = 3x + 2 \), the y-intercept \( b \) was calculated to be 2, meaning the graph crosses the y-axis at the point (0, 2). This point is essential because it provides the anchor for the line as it stretches across the graph according to its slope.
Understanding the y-intercept helps set the foundation for plotting the graph and ensuring accuracy as you draw lines and predict additional points. Plotting the y-intercept on the graph means you can easily sketch or verify the behavior of the entire function with simplicity and precision.