Problem 54 - Algebra 2

Question

Find the slope and the intercepts of each line. $$ f(x)=\frac{2}{3} x+4 $$

Step-by-Step Solution

Answer

The slope is \(\frac{2}{3}\) and the \(y\)-intercept is \(4\). The \(x\)-intercept is \( -6\).

Step 1: Identify the Slope

The equation given is in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope. For the given function \(f(x) = \frac{2}{3}x + 4\), the slope \(m\) is \(\frac{2}{3}\).

Step 2: Identify the Y-intercept

From the slope-intercept form, the \(y\)-intercept is given by \(b\), the constant in the equation. For \(f(x) = \frac{2}{3}x + 4\), the \(y\)-intercept is \(4\), which means the line crosses the \(y\)-axis at \(y = 4\).

Step 3: Calculate the X-intercept

To find the \(x\)-intercept, set the output \(f(x)\) to zero and solve for \(x\). \[0 = \frac{2}{3}x + 4\] Subtract \(4\) from both sides: \[\frac{2}{3}x = -4\] Multiply both sides by \(\frac{3}{2}\) to solve for \(x\): \[x = -4 * \frac{3}{2} = -6\] So the \(x\)-intercept is \(x = -6\).

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in interpreting the rate at which one quantity changes with respect to another. In simple terms, the slope indicates the steepness and direction of a line on a graph. The equation of a line in slope-intercept form is expressed as
\( y = mx + b \)
where \( m \) represents the slope. If the slope is positive, the line rises from left to right, and if negative, it falls from left to right. A slope of zero indicates a horizontal line, and undefined slope (division by zero) corresponds to a vertical line. To find the slope from the given equation,
\( f(x) = \frac{2}{3}x + 4 \),
we can see that the coefficient of \(x\) is \(\frac{2}{3}\), which is the slope. This value means that for each unit increase in \(x\), the value of \(f(x)\) or \(y\) increases by \(\frac{2}{3}\) units.

Y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. This occurs when the value of \(x\) is zero. In the slope-intercept form,
\( y = mx + b \),
\(b\) is the y-intercept. It represents the value of \(y\) when \(x\) is zero. For instance, in the equation
\( f(x) = \frac{2}{3}x + 4 \),
the y-intercept is \(4\). It tells us that the point (0, 4) is on the line, providing a starting point for drawing or extending the graph of the line. The y-intercept is often used to interpret data in various contexts, such as in business to determine fixed costs or in science to understand initial conditions.

X-intercept

Conversely, the x-intercept is where the line crosses the x-axis. Determining this point involves setting the \(y\) value to zero and solving the equation for \(x\). In our example,
\( f(x) = \frac{2}{3}x + 4 \),
finding the x-intercept requires that we solve the equation \(0 = \frac{2}{3}x + 4\). After rearranging terms and solving for \(x\), we find that the x-intercept is \(-6\). This calculation reveals that the line will cross the x-axis at the point (-6, 0). The x-intercept is particularly useful in break-even analysis in economics or finding the roots of functions in mathematics.

Linear Equations

Linear equations form the foundation for much of algebra and serve as a starting point for calculus and other higher-level mathematics. These equations graph as straight lines and can be written in various forms, with the slope-intercept form, \( y = mx + b \), being one of the most commonly used. It clearly shows the slope and y-intercept, making it easy to graph the line or to interpret its behavior. Lines described by linear equations can represent simple relationships between variables, such as cost in relation to production level or distance in relation to time. When given in standard form, \( Ax + By = C \), one may rearrange the equation to the slope-intercept form to more easily identify the slope and intercepts, which we effectively used in our example to describe the behavior of the function \( f(x) \).