Math Problem Statement

Find the slope, y-intercept, and equation of the line from the graph.

Solution

In this graph, we are tasked with finding the slope, y-intercept, and equation of the line.

1. Finding the Slope:

The slope of a line is given by the formula: m=change in ychange in x=y2y1x2x1m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} From the graph, we can identify two points on the line. Let's say the points are:

  • (0,2)(0, 2) where the line intersects the y-axis.
  • (4,2)(4, -2) another point on the line.

Now, apply the slope formula: m=2240=44=1m = \frac{-2 - 2}{4 - 0} = \frac{-4}{4} = -1 So, the slope (m) is -1.

2. Finding the Y-intercept:

The y-intercept is the point where the line crosses the y-axis. From the graph, it's clear that the line crosses the y-axis at (0,2)(0, 2). Thus, the y-intercept is 22.

3. Finding the Equation:

The equation of a line in slope-intercept form is: y=mx+by = mx + b Where mm is the slope and bb is the y-intercept.

Substituting the values: y=1x+2ory=x+2y = -1x + 2 \quad \text{or} \quad y = -x + 2 Thus, the equation of the line is: y=x+2y = -x + 2

Summary:

  • Slope (m): 1-1
  • Y-intercept (b): 22
  • Equation: y=x+2y = -x + 2

Would you like more details or have any questions about this? Here are 5 related questions to help expand on this:

  1. How do you calculate the slope if given two arbitrary points?
  2. What does the slope of 1-1 indicate about the line's steepness and direction?
  3. How would you find the x-intercept of this line?
  4. Can you convert the equation y=x+2y = -x + 2 into standard form?
  5. How does the graph of this line change if the slope or y-intercept were different?

Tip: When plotting a line from an equation, start by plotting the y-intercept, then use the slope to find another point.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Slope
Y-Intercept
Equation of a Line

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Slope-intercept form: y = mx + b

Theorems

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Suitable Grade Level

Grades 7-8