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Chapter 1: Problem 13

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. To graph this function, plot the given points (1, 3) and (2, 5) on a coordinate plane and draw a straight line through them. The line will have a slope of 2, meaning it goes up by 2 units for every 1 unit it moves horizontally to the right.

Step by step solution

01

Find the slope (m)

To find the slope (m) of the linear function, we can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points given. Plugging in the given points, we get: m = (5 - 3) / (2 - 1) = 2 / 1 = 2 So the slope of the function is 2.
02

Find the y-intercept (b)

Now that we have the slope, we can find the y-intercept (b) by substituting one of the given points into the slope-intercept form of a linear equation (y = mx + b). Let's use the point (1, 3): 3 = 2(1) + b Solving for b, we get: b = 3 - 2 = 1 So the y-intercept of the function is 1.
03

Write the equation of the linear function

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

Graph the function

To graph the linear function y = 2x + 1, we start by plotting the two given points (1, 3) and (2, 5) on a coordinate plane. Then, we draw a straight line through these points. This line represents the graph of the linear function. The line will have a slope of 2, meaning it goes up by 2 units for every 1 unit it moves horizontally to the right.

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Key Concepts

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

Slope-Intercept Form
A linear equation can often be easily expressed through the slope-intercept form: \[ y = mx + b \] Here, \( m \) represents the slope of the line, and \( b \) indicates the y-intercept. The slope \( m \) tells us how steeply the line rises or falls as it moves horizontally across the graph. If the slope is positive, the line slopes upwards. If negative, it slopes downwards. The y-intercept \( b \) is the point where the line crosses the y-axis, which means the value of \( y \) when \( x = 0 \). For example, in the equation \( y = 2x + 1 \):
  • The slope \( m = 2 \) shows that for every step we move right, \( y \) increases by 2 units.
  • The y-intercept \( b = 1 \) tells us the line crosses the y-axis at (0,1).
Using this form makes it straightforward to analyze and graph linear functions.
Graphing Linear Equations
Graphing linear equations is a powerful way to visually understand and interpret data. To graph a linear equation like \( y = 2x + 1 \), follow these simple steps:
  • Start with the y-intercept: Plot the y-intercept \( (0, b) \). Here, that's the point (0,1).

  • Use the slope: From the y-intercept, use the slope \( m \). For \( m = 2 \), move up 2 units and right 1 unit from the y-intercept to find another point on the line, such as (1,3).

  • Extend the line: Connect these points with a straight line. Extend it in both directions to cover more of the graph.
This creates a graphical representation of the linear function. Every point on the line satisfies the given equation. Remember, the slope sets the angle, and the y-intercept sets the starting point. It's amazing how well these simple steps define the entire line!
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, helps us understand geometric figures using a coordinate system. It unites algebra and geometry by describing geometric shapes in numerical terms. In our example, the coordinates (1,3) and (2,5) are used to find the line's slope and equation. Here's how they work together:
  • Coordinates: Each point on the graph is in the form \((x, y)\), linking algebraic expressions to geometrical locations.
  • Slope calculation: By taking the change in \( y \) coordinates (the rise) over the change in \( x \) coordinates (the run), we determine how the line ascends or descends, which gives the slope.
  • Graphing: Once the linear equation is put together from the coordinates, we transform abstract numbers into a visual line on the Cartesian plane.
Coordinate geometry makes it easier to solve complex geometric problems and to visualize equations as paths on a plane.

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