Problem 37 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(--\frac{1}{2}\) and \(y\) -intercept \(-4\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y = 8x + 4\). The slope-intercept form of the line is \(y = 8x - 4\).

Step 1: Determine Points

The provided intercepts are the points where the line crosses the x and y axis. Therefore these intercepts are actually points on the line and can be represented in the form (x,y). The given x-intercept is \(-\frac{1}{2}\), so the corresponding point on the line is \(-\frac{1}{2},0\). The y-intercept is -4, so the corresponding point on the line is \(0,-4\).

Step 2: Calculate Slope

Given two points \(x_1,y_1\) and \(x_2,y_2\), the slope of the line that passes through them can be determined using the formula \(m=\(\frac{y_2-y_1}{x_2-x_1}\). Applying this to our points, we get: \(m=\(\frac{-4-0}{0-\(-\frac{1}{2}\)}\). Simplifying this, the slope is \(m=8\).

Step 3: Apply Point-Slope Form

The point-slope form of a line is \(y-y_1=m(x-x_1)\). Using the point \(-\frac{1}{2},0\) and the slope \(8\), the point-slope form of the line is \(y-0=8(x-(-\frac{1}{2})\), after simplifying it becomes \(y = 8x + 4\).

Step 4: Apply Slope-Intercept Form

The slope-intercept form of a line is \(y=mx+b\). The y-intercept (b) is given. The calculated value of the slope (m) was \(8\). Substitute these values into the slope-intercept form, we have \(y = 8x - 4\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is particularly useful when you know one point on a line and the slope of the line. The formula is expressed as:
\[ y - y_1 = m(x - x_1) \]
Where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. This form can help you quickly determine the equation if you know the initial point and how steep the line is.
To illustrate, using the point \(-\frac{1}{2},0\) and the calculated slope \(8\), you plug these values into the point-slope formula:

• Start: \( y - 0 = 8(x - (-\frac{1}{2})) \)
• Simplify: \( y = 8x + 4 \)

This is a straightforward way of seeing how changes in \( x \) affect changes in \( y \) on your line.

Slope-Intercept Form

The slope-intercept form is one of the most commonly used forms of linear equations because it gives you a clear visual of the line's slope and the \( y \)-intercept immediately. It is written as:
\[ y = mx + b \]
Here, \( m \) represents the slope, and \( b \) is the \( y \)-intercept. This form allows you to graph the line easily since you can begin at the \( y \)-intercept and use the slope to find additional points.
Now, using the earlier results:

• Slope: \( m = 8 \)
• \( y \)-intercept: \( b = -4 \)

Substitute these into the formula:

• Equation: \( y = 8x - 4 \)

With this form, identifying key aspects of the line and sketching it becomes much simpler.

Intercepts

Intercepts are the points where the line crosses the axes. Understanding these can give you a good starting point for plotting the line.
- **\(x\)-Intercept**: This is where the line crosses the \(x\)-axis. The \(y\)-value at this point is 0. You can denote the \(x\)-intercept as \((-\frac{1}{2}, 0)\).
- **\(y\)-Intercept**: This is where the line crosses the \(y\)-axis, and the \(x\)-value here is 0. This point is \((0, -4)\).
By knowing both intercepts, you not only confirm the positions where the line intersects the axes, but you also can quickly determine important characteristics, such as the slope, which will make drawing the line easier.

Slope Calculation

Calculating the slope is crucial in understanding the relationship between the variables of a line. The slope (\(m\)) quantifies the change in \(y\) with respect to each unit change in \(x\).
The formula for slope given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
With our points:\( (-\frac{1}{2}, 0) \) and \((0, -4)\), plug them into the formula:

• Calculate: \( m = \frac{-4 - 0}{0 - (-\frac{1}{2})} \)
• Simplify: \( m = 8 \)

This slope tells you that for every unit you move to the right on the \(x\)-axis, the line moves up by 8 units. Understanding slope gives you a sense of direction and steepness essential for sketching the graph of the line.