Math Problem Statement

What is the slope of the line graphed above? Give your answer as a reduced fraction. Identify the y-intercept. Write it as a point. Write an equation for the line in slope-intercept form.

Solution

Let's solve the problem step by step based on the given graph:

1. Find the Slope

The slope (mm) of a line is calculated as: m=riserunm = \frac{\text{rise}}{\text{run}}

Looking at the graph:

  • Take two clear points on the line, e.g., (4,2)(-4, -2) and (2,4)(2, 4).
  • Calculate the rise (change in yy): 4(2)=64 - (-2) = 6.
  • Calculate the run (change in xx): 2(4)=62 - (-4) = 6.

Thus, the slope is: m=66=1m = \frac{6}{6} = 1

2. Identify the y-Intercept

The yy-intercept (bb) is the point where the line crosses the yy-axis. From the graph, the line crosses the yy-axis at y=2y = 2. Therefore, the yy-intercept is: (0,2)(0, 2)

3. Write the Equation of the Line

The slope-intercept form of a line is: y=mx+by = mx + b

Substitute m=1m = 1 and b=2b = 2: y=1x+2or simplyy=x+2y = 1x + 2 \quad \text{or simply} \quad y = x + 2

Final Answers:

  1. Slope: 11
  2. y-Intercept: (0,2)(0, 2)
  3. Equation: y=x+2y = x + 2

Would you like further explanation on any of these steps? Here are some related questions to explore:

  1. What does a slope of 11 signify geometrically?
  2. How do you calculate slope with negative coordinates?
  3. Can you identify other points on this line using its equation?
  4. How does changing the yy-intercept affect the graph?
  5. What happens to the equation if the slope is steeper (e.g., m=2m = 2)?

Tip: Always verify your slope and intercept by checking with at least two points from the graph!

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Math Problem Analysis

Mathematical Concepts

Slope
Y-Intercept
Linear Equations

Formulas

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

Theorems

Properties of linear equations

Suitable Grade Level

Grades 7-10