Problem 38 - Algebra 1

Question

Graph the function. $$ f(x)=4 x+1 $$

Step-by-Step Solution

Answer

The function \( f(x) = 4x + 1 \) is graphed by first plotting the points (0,1) and (1,5), and then drawing a line through these points.

Step 1: Identify the Slope and Y-Intercept

The given function is \( f(x) = 4x + 1 \). When comparing this with the slope-intercept form \( y = mx + c \), we find that the slope 'm' is 4 and the y-intercept 'c' is 1. This means, for every unit increase in 'x', the value of 'f(x)' increases by 4 units, and when 'x' equals 0, 'f(x)' (or 'y') equals 1.

Step 2: Plot the Y-Intercept

We start by plotting the y-intercept on the graph, which is the point (0,1).

Step 3: Use the Slope to Find Another Point

Now utilize the slope to find another point on the line. Given the slope is 4, we may go 1 unit to the right on 'x' and 4 units up on 'f(x)', resulting in a new point (1,5). Plot this point on the graph.

Step 4: Draw the Line

Finally, a straight line is drawn through the two plotted points, thus graphing the linear function \( f(x) = 4x + 1 \).

Key Concepts

Graphing Functions

Graphing functions provides a visual representation of relationships between variables. For linear functions, like the one in our exercise, the graph will always be a straight line.
To graph a linear function, we start by identifying specific points through which the line will pass. The basic strategy involves using the function's slope and y-intercept to determine these points.

• The first step is identifying the y-intercept, which is where the line crosses the y-axis. This can be easily found as it is the constant in the function when it is in slope-intercept form.
• The next step is using the slope, which tells us how steep the line is, to find another point. Once we have two points, we can draw a straight line through them to complete the graph.

Graphing is a fundamental skill in understanding how linear functions behave and how changes in equations affect the graphical representation.

Slope-Intercept Form

The slope-intercept form is a popular and highly useful way to represent linear functions. Denoted as \( y = mx + c \), it provides immediate insights into the structure and characteristics of the function.
The function in slope-intercept form gives us two key pieces of information:

• The slope \( m \), which indicates how the function values change with respect to changes in the \( x \)-values. It represents the steepness and direction of the line.


• The y-intercept \( c \), which is the point where the line crosses the y-axis. This value indicates the output when \( x = 0 \).

This form is especially beneficial for quickly graphing functions and understanding their basic properties, allowing students to visualize changes in linear equations effectively.

Slope and Y-Intercept

Understanding both the slope and y-intercept is crucial for manipulating and graphing linear functions effectively.
The slope is a measure of how fast or slow the function value (or \( y \)-value) rises or falls as the \( x \)-value increases. In the equation \( f(x) = 4x + 1 \), the slope \( m \) is 4, meaning that for every 1 unit increase in \( x \), \( y \) increases by 4 units. A larger slope leads to a steeper line, while a smaller slope yields a flatter one.

The y-intercept, indicated as \( c \) in the function \( y = mx + c \), is the output value when input \( (x) \) is zero. For the function in the exercise, it is 1, which means the line will cross the y-axis at (0,1).

• To graph, start at the y-intercept on the y-axis.
• Use the slope to move to the next point. For each unit you move right along the x-axis, move up according to the slope value.

These concepts together facilitate the understanding of how linear functions behave and make it easier to predict their graphical features.