Problem 59 - Algebra 1: Concepts and Skills

Question

Graph the function. (Lesson 4.8) $$ h(x)=4 x-4 $$

Step-by-Step Solution

Answer

The y-intercept of the function \(h(x) = 4x - 4\) is -4. The slope of the function is 4. Thus, the function is graphed as a straight line starting at -4 on the y-axis and going upwards at an angle that corresponds to a 'rise over run' of 4.

Step 1: Identify the y-intercept

To find the y-intercept, plug in \(x = 0\) into the function. \[h(0)=4*0 - 4 = -4\] therefore, the y-intercept is -4.

Step 2: Identify the slope

The slope of the function is the coefficient of \(x\), which is 4 in our function. This means, for each 1 unit increase in \(x\), \(y\) increases by 4 units.

Step 3: Draw the graph

Start by marking the y-intercept (-4) on the vertical axis. Then, using the slope, move up 4 units and 1 unit to the right from the y-intercept to find the next point for \(x = 1\), continue this for a couple more points. Once the points are plotted, draw a straight line through them. The graph of the function is this drawn line.

Key Concepts

Understanding the Y-Intercept

The y-intercept is a crucial component when graphing linear functions. This is the point where the line crosses the y-axis on the graph. To find it, set the variable \(x\) to zero in the equation, as the y-intercept occurs where the line meets the vertical axis (y-axis), at \(x = 0\).
For example, in the function \(h(x) = 4x - 4\), substituting \(x = 0\) allows us to calculate the y-intercept: \\[h(0) = 4(0) - 4 = -4\]
Thus, the y-intercept is \(-4\). This implies that the first point to plot on the graph is on the y-axis at \( (0, -4) \). Understanding how to find and plot the y-intercept is a key step in sketching linear equations on a graph.

Learning About the Slope

The slope is another fundamental aspect of graphing linear equations. It measures the steepness and direction of the line. In simple terms, the slope indicates how much the output \(y\) changes as the input \(x\) changes.
In the equation \( y = mx + b \), the \(m\) represents the slope.

• If the slope is positive, the line rises as it moves from left to right.
• If it is negative, the line falls as it moves from left to right.
• A larger slope number means a steeper line.

In our example, \(h(x) = 4x - 4\), the slope is \(4\). This means for each increase of 1 unit in \(x\), \(y\) will increase by 4 units. Therefore, from the y-intercept at \(-4\) (point \( (0, -4) \)), moving up 4 units and right 1 unit gives the next point on the line at \( (1, 0) \). This concept of slope helps in drawing a precise line on the graph that represents the equation.

The World of Linear Equations

Linear equations form the backbone of many algebraic concepts and real-world applications. A linear equation is any equation that can be written in the standard form \(y = mx + b\), where \(m\) and \(b\) are constants.
These equations produce straight lines when plotted on a graph. The term "linear" indicates the graphing of these equations results in a line.

• \(y = mx + b\) where \(m\) stands for the slope and \(b\) is the y-intercept.
• These components determine the line’s angle (slope) and starting point (y-intercept) on the graph.
• Simplifying an equation into this form allows easier graph analysis.

By understanding linear equations, you can better predict how changes in \(x\) will affect \(y\). This foundational knowledge is not only practical for graphing but also essential for solving real-world problems involving ratios and proportional reasoning.