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Chapter 1: Problem 15

Graph of a linear function Find and graph the linear function that passes through the points (1,3) and (2,5)

Short Answer

Expert verified
Answer: The equation of the linear function that passes through the points (1,3) and (2,5) is y = 2x + 1. The slope (m) is 2, and the y-intercept (b) is 1.

Step by step solution

01

Calculate the Slope

To determine the slope (m), we can use the formula m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the given points. In this case, x1 = 1, y1 = 3, x2 = 2, and y2 = 5. Substituting these values into the formula, we get: m = (5 - 3) / (2 - 1) = 2 / 1 = 2 The slope of the linear function is 2.
02

Find the Y-intercept

To find the y-intercept (b), we can use the equation of a line in slope-intercept form, y = mx + b, where m is the slope and (x, y) is a point on the line. We've already determined that m = 2. We can use one of the given points, say (1, 3), to find b: 3 = 2(1) + b 3 = 2 + b b = 1 The y-intercept of the linear function is 1.
03

Write the Equation of the Line

Now that we have the slope (m) and y-intercept (b), we can write the equation of the linear function. In slope-intercept form, the equation is y = mx + b: y = 2x + 1 The equation of the linear function that passes through the points (1,3) and (2,5) is y = 2x + 1.
04

Graph the Linear Function

To graph the linear function y = 2x + 1, follow these steps: 1. Begin by plotting the y-intercept (b) on the y-axis; in this case, it's at a point (0,1). 2. Use the slope (m) as rise/run to find the next point. Since the slope is 2, this means a rise of 2 and a run of 1. From the y-intercept point (0,1), move up 2 units and 1 unit to the right on the graph to find the next point (1,3). 3. To verify the work, plot the other given point (2,5) on the graph. 4. Draw a straight line through the points (0,1), (1,3), and (2,5). This represents the linear function y = 2x + 1. The graph of the linear function that passes through the points (1,3) and (2,5) is a straight line with slope 2 and y-intercept 1, represented by the equation y = 2x + 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The slope of a line tells us how steep the line is. It's a measure of how much the line rises or falls as it moves from left to right across a graph. To find the slope between two points, we use the formula
  • \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the points.
In this problem, we are given two points: (1,3) and (2,5). Plug these into the formula to get:
  • \( m = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2 \)
This means the line rises 2 units for every 1 unit it runs to the right. The slope is 2, indicating a moderately steep line. Understanding slope is crucial when learning how graphs represent relationships in math and real-life scenarios.
Y-intercept
The y-intercept is the point where the line crosses the y-axis. It's important because it helps us set the position of the line on a graph. In any linear function written in slope-intercept form, \(y = mx + b\), the y-intercept is \(b\).
To find the y-intercept when given a slope and a point, use the formula by substituting the values you have. Here, with a slope \(m = 2\) and using the point (1, 3):
  • \(3 = 2 \cdot 1 + b\)
  • \(3 = 2 + b\)
  • \(b = 1\)
So, the y-intercept is 1. This means the graph crosses the y-axis at (0,1). Y-intercepts provide quick information about starting values or initial conditions in many contexts.
Slope-Intercept Form
The slope-intercept form of a linear equation is a user-friendly way to express a line's equation. It is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This form is incredibly useful because it instantly tells us the slope and y-intercept of the line without needing additional calculations. It's straightforward to graph, as you start by plotting the y-intercept, then use the slope to find other points.
For the current problem, the slope is 2 and the y-intercept is 1, leading us to the equation:
  • \(y = 2x + 1\)
With this equation, anytime we have a value for \(x\), we can find the corresponding \(y\). It's perfect for quickly graphing or understanding linear relationships.
Plotting Points
Plotting points on a graph is like connecting the dots to form a picture of the equation. It helps visualize how the equation behaves and its direction.
  • Start with the y-intercept (0,1). Plot this on the y-axis.
  • Next, use the slope to find another point. With a slope of 2, move up 2 units and right 1 unit from the y-intercept to get to (1,3).
  • Check another known point, like (2,5), to ensure your graph's accuracy.
When you have at least two points, draw a line through them. This connects the dots and illustrates the path of the linear function. Remember, lines in a linear function extend infinitely in both directions, so continue the line beyond the plotted points to fully represent the graph.

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