Problem 46 - Algebra 2

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Then graph the equation. \(f(x)=4 x-1\)

Step-by-Step Solution

Answer

y-intercept: (0, -1); x-intercept: \(\left(\frac{1}{4}, 0\right)\).

Step 1: Find the y-intercept

The y-intercept is the point where the graph of the equation crosses the y-axis. At this point, the value of x is 0. Substitute x = 0 in the equation: \[ f(x) = 4(0) - 1 = -1 \]Therefore, the y-intercept is the point (0, -1).

Step 2: Find the x-intercept

The x-intercept is the point where the graph of the equation crosses the x-axis. At this point, the value of f(x) is 0. Set f(x) to 0 and solve for x:\[ 0 = 4x - 1 \]Add 1 to both sides:\[ 1 = 4x \]Divide both sides by 4:\[ x = \frac{1}{4} \]Thus, the x-intercept is the point \(\left(\frac{1}{4}, 0\right)\).

Step 3: Graph the equation

Plot the intercepts on a coordinate plane. Start with the y-intercept (0, -1) and mark it on the y-axis. Then plot the x-intercept \(\left(\frac{1}{4}, 0\right)\) on the x-axis. Draw a straight line through these two points to represent the graph of the equation f(x) = 4x - 1. This line has a constant slope of 4, which confirms that it is a straight line.

Key Concepts

Understanding the X-Intercept

The x-intercept is a crucial aspect of graphing linear equations. It is the point on the graph where the line crosses the x-axis. At this specific location, the value of y is always zero since the point lies precisely on the x-axis.
To find the x-intercept, we set the function equal to zero and solve for x. For the function \(f(x)=4x-1\), we substitute \(f(x)\) with zero, leading us to the equation \(0=4x-1\). By solving this equation, we add 1 to both sides to get \(1=4x\), and then divide by 4 to isolate x, which gives us \(x=\frac{1}{4}\).
Therefore, the x-intercept is the point \(\left(\frac{1}{4}, 0\right)\). This is fundamental for graphing because it provides a starting point on the x-plane.

Exploring the Y-Intercept

The y-intercept is the point where the line crosses the y-axis, a key detail when sketching a linear equation. At this point, the value of x is always zero since the point lies directly on the y-axis.
To find the y-intercept in the function \(f(x)=4x-1\), substitute x with zero, which gives the equation \(f(x)=4(0)-1=-1\). This means the y-intercept is at the point \((0,-1)\).
Remember, knowing the y-intercept helps to orient the line correctly on the graph, as it is the initial point where the line begins vertically. It can also help in understanding the equation's behavior at the start of the graph.

Graphing Linear Equations

Graphing linear equations is a process of making a visual representation on a coordinate plane. This graph portrays how the variables relate. We do this by plotting important points, like the x-intercept and y-intercept, and then drawing a line through them.
To graph \(f(x)=4x-1\), start by marking the intercepts. Plot the y-intercept \((0,-1)\) on the y-axis. Then, plot the x-intercept \(\left(\frac{1}{4}, 0\right)\) on the x-axis. These two points serve as crucial anchors for the line.

• Mark these points clearly on your graph.
• Use a ruler or straightedge to draw a straight line through these points.
• This line represents the equation \(f(x)=4x-1\), which has a consistent slope of 4.

This process of graphing helps to visually see the linear relationship, confirming how the equation behaves across the axes.