Math Problem Statement
Select the correct answer. A linear function on a coordinate plane passes through (minus 3, 3), (0, 2), and (3, 1)
In the function above, the slope will be multiplied by -9, and the y-value of the y-intercept will be increased by 2 units. Which of the following graphs best represents the new function?
A linear function on a coordinate plane passes through (minus 1, 3), (0, 0), and (1, minus 3) W. The graph shows a linear function passes through (0, 4), (1.2, 0), and (3, minus 5) X. The graph shows a linear function passes through (0, 4), (minus 1.2, 0), and (minus 3, minus 5) Y. A linear function on a coordinate plane passes through (1, 3), (0, 0), and (minus 1, minus 3) Z.
A. W B. X C. Z D. Y
Solution
To solve this, let’s break down the steps clearly:
Step 1: Find the slope of the original function
The original function passes through points , , and .
We use the formula for the slope between two points: Let’s use the points and :
So, the slope of the original line is .
Step 2: Apply the changes described in the problem
-
Multiply the slope by -9:
-
Increase the y-value of the y-intercept by 2 units:
The original y-intercept can be found by using the point (since at , ).After increasing the y-intercept by 2 units, the new y-intercept is .
Step 3: Equation of the new function
Now that we have the new slope of and the new y-intercept of , the equation of the new line is:
Step 4: Look for the graph that matches this new line
Now, let's look for a graph that has a slope of 3 and a y-intercept of 4.
The graph options:
- W shows the line passing through , but the slope is not consistent with 3.
- X shows the line passing through , but the slope is inconsistent with 3.
- Y shows a line passing through , which is incorrect.
- Z shows a line passing through and the slope seems to match.
Based on this analysis, the correct answer is B. X.
Would you like a more detailed explanation or help with other questions?
Follow-up questions:
- How do you determine the slope from two points on a line?
- What happens to the graph if the slope is multiplied by a negative number?
- How can you find the y-intercept from a given point on the line?
- Why is the equation of a line in the form ?
- What are the key features to look for when comparing linear functions on a graph?
Tip:
When finding the equation of a line, always remember to calculate the slope first and then use a known point to find the y-intercept.
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Math Problem Analysis
Mathematical Concepts
Linear Functions
Slope
Y-intercept
Graphing Linear Equations
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + b
Theorems
Slope of a line
Linear equation transformations
Suitable Grade Level
Grades 7-9
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