Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) $$\text { Through }(-4,3), m=0.75$$

Step-by-Step Solution

Answer

The equation of the line is \( y = 0.75x + 6 \).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Identify Given Information

From the problem, we know that the slope \(m = 0.75\) and the line passes through the point \((-4, 3)\).

Step 3: Substitute into Slope-Intercept Equation

Plug the point and slope into the equation to solve for \(b\): \( 3 = 0.75(-4) + b \).

Step 4: Solve for the Y-Intercept \(b\)

Calculate \(0.75 \times -4 = -3\), so \(3 = -3 + b\). Add \(3\) to both sides to find \(b = 6\).

Step 5: Write the Final Equation

Substitute \(m = 0.75\) and \(b = 6\) back into the slope-intercept form: \( y = 0.75x + 6 \).

Key Concepts

Linear Equations

A linear equation represents a straight line on a graph. The general form of a linear equation is given by

• Standard Form: \[ Ax + By = C \] where \(A\), \(B\), and \(C\) are constants.
• Slope-Intercept Form: \[ y = mx + b \] where \(m\) is the slope, and \(b\) is the y-intercept.

These equations are important for modeling relationships where one variable changes consistently with another.
For instance, predicting cost over time, where time is the independent variable \(x\) and cost is the dependent variable \(y\).
This consistent change makes a straight line, which is easily identifiable in graphs.

Slope

The slope of a line reflects how steeply it rises or falls on a graph. It measures the rate of change between the variables. For the slope \(m\):

• A positive slope indicates the line is rising or going upwards. For example, \(m = 0.75\) means for each unit increase in \(x\), \(y\) increases by 0.75.
• A negative slope means the line goes downwards.
• If the slope is zero, the line is horizontal, indicating no change in \(y\) regardless of \(x\).

We typically find the slope by taking two points on the line.

• Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

It tells us the exact steepness and the direction the line goes on a graph.

Y-Intercept

The y-intercept is where the line crosses the y-axis. It's denoted by \(b\) in the slope-intercept equation \(y = mx + b\). This point is particularly important when you need to quickly determine if a line intersects a particular y-value. In our example, the y-intercept \(b = 6\).

• To find it, solve for \(b\) using known values of \(x\) and \(y\) from a point on the line. Example: \[ 3 = 0.75 \times (-4) + b \], solving gives \(b = 6\).

This concept is vital for real-world scenarios, providing answers to questions like initial costs before additional inputs are added.

Coordinates

Coordinates are pairs of numbers \((x, y)\) that define a point's position on a graph. They are essential for plotting and understanding linear equations. The first number, \(x\), is the horizontal position, while \(y\) is the vertical position.
Using coordinates, you can:

• Plot points to visualize lines and curves on a coordinate plane.
• Calculate the slope by using differences between coordinates of two points.
• Determine if a point lies on a certain line.

In our example, the point \((-4, 3)\) helps to find where the graph of \(y = 0.75x + 6\) should lie.
This knowledge assists in comprehending how every input \(x\) ties to an output \(y\) on the graph of an equation.