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 (3,3)(-3, 3), (0,2)(0, 2), and (3,1)(3, 1).

We use the formula for the slope between two points: slope=y2y1x2x1\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} Let’s use the points (0,2)(0, 2) and (3,1)(3, 1): slope=1230=13\text{slope} = \frac{1 - 2}{3 - 0} = \frac{-1}{3}

So, the slope of the original line is 13-\frac{1}{3}.

Step 2: Apply the changes described in the problem

  • Multiply the slope by -9:
    New slope=13×9=3\text{New slope} = -\frac{1}{3} \times -9 = 3

  • Increase the y-value of the y-intercept by 2 units:
    The original y-intercept can be found by using the point (0,2)(0, 2) (since at x=0x = 0, y=2y = 2).

    After increasing the y-intercept by 2 units, the new y-intercept is 2+2=42 + 2 = 4.

Step 3: Equation of the new function

Now that we have the new slope of 33 and the new y-intercept of 44, the equation of the new line is: y=3x+4y = 3x + 4

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 (0,4)(0, 4), but the slope is not consistent with 3.
  • X shows the line passing through (0,4)(0, 4), but the slope is inconsistent with 3.
  • Y shows a line passing through (0,0)(0, 0), which is incorrect.
  • Z shows a line passing through (0,4)(0, 4) 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:

  1. How do you determine the slope from two points on a line?
  2. What happens to the graph if the slope is multiplied by a negative number?
  3. How can you find the y-intercept from a given point on the line?
  4. Why is the equation of a line in the form y=mx+by = mx + b?
  5. 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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