Linear Functions – Graphing Lines Given Y-intercept and an Ordered Pair Workshee

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Linear Functions – Graphing Lines Given Y-intercept and an Ordered Pair Worksheet – Sketch the Line and Write the Equation for the Line Given Two Ordered Pairs

Introduction

In this worksheet, we will further explore linear functions and delve into graphing lines given the y-intercept and two ordered pairs. Linear functions play a crucial role in mathematics and have wide applications in various disciplines. Understanding how to graph lines accurately and write their equations is essential for analyzing data and solving real-world problems.

Understanding Linear Functions and Equations

Before we proceed, let’s review the basics of linear functions and equations. A linear function is a mathematical representation of a straight line on a graph. It can be expressed in the form:

y = mx + b

In this equation, y represents the dependent variable, x represents the independent variable, m represents the slope of the line, and b represents the y-intercept. The slope determines the steepness of the line, while the y-intercept represents the point where the line intersects the y-axis.

Graphing Lines Given the Y-intercept and Two Ordered Pairs

Now, let’s discuss how to graph lines when given the y-intercept and two ordered pairs. This method allows us to sketch a line accurately on a coordinate plane using more information, resulting in a more precise representation of the line.

Step 1: Identify the Y-intercept

The y-intercept is the point where the line intersects the y-axis. It is represented by the coordinate (0, b), where b is the y-intercept value provided in the problem. This point will serve as our starting point for graphing the line.

Step 2: Calculate the Slope

To calculate the slope of the line, we use the two given ordered pairs, (x1, y1) and (x2, y2). The slope can be determined using the formula:

m = (y2 – y1) / (x2 – x1)

Substitute the coordinates of the ordered pairs into the formula to find the slope value.

Step 3: Plot the Second Ordered Pair

Using the slope obtained in the previous step, we can now find a second point on the line. Starting from the y-intercept, use the slope to determine the direction and distance to move horizontally and vertically. Plot this second ordered pair on the coordinate plane.

Step 4: Connect the Points and Determine the Equation

With both ordered pairs plotted, draw a straight line passing through them. Ensure the line extends beyond the plotted points to represent its infinite nature. This line represents the graph of the linear function.

To determine the equation of the line, we can use the point-slope form:

y – y1 = m(x – x1)

Substitute one of the ordered pairs and the slope into this equation to find the final equation of the line.

Practice Worksheet

To reinforce the concepts we have discussed, let’s work through a practice worksheet that will guide you in graphing lines and writing their equations given the y-intercept and two ordered pairs.

1. Graph the line with a y-intercept of 3 and the ordered pairs (-2, 1) and (2, 7). Write the equation of the line.
2. Sketch the line given a y-intercept of -4 and the ordered pairs (3, 5) and (-1, -3). Determine the equation of the line.
3. Plot the line with a y-intercept of 2 and the ordered pairs (0, 2) and (4, 8). Find the equation of the line.
4. Graph the line given a y-intercept