Math Problem Statement

What is the x-intercept of the graph of this function?

Solution

To find the x-intercept of the graph of the linear function shown in the table, we first need to establish the equation of the line by determining the slope and y-intercept, then solve for the x-intercept.

Step 1: Find the slope (m)

The slope mm of a line is given by: m=y2y1x2x1m = \frac{{y_2 - y_1}}{{x_2 - x_1}} We can take any two points from the table. Let's use the points (-3, 18) and (-2, 16): m=16182(3)=21=2m = \frac{{16 - 18}}{{-2 - (-3)}} = \frac{{-2}}{{1}} = -2 Thus, the slope is m=2m = -2.

Step 2: Use the point-slope form to find the equation of the line

We can now use the point-slope form of the equation of a line, which is: yy1=m(xx1)y - y_1 = m(x - x_1) Substituting m=2m = -2 and the point (-3, 18): y18=2(x+3)y - 18 = -2(x + 3) Simplifying: y18=2x6y - 18 = -2x - 6 y=2x+12y = -2x + 12 So the equation of the line is y=2x+12y = -2x + 12.

Step 3: Find the x-intercept

The x-intercept occurs when y=0y = 0. Substituting y=0y = 0 into the equation: 0=2x+120 = -2x + 12 Solving for xx: 2x=122x = 12 x=6x = 6

Thus, the x-intercept is x=6x = 6.

The correct answer is 6.

Would you like further explanation or details?


Here are five related questions to expand your understanding:

  1. What is the significance of the slope in a linear function?
  2. How can you quickly identify the y-intercept from the equation of a line?
  3. How do changes in slope affect the steepness of a line?
  4. What are the differences between x-intercepts and y-intercepts?
  5. How would you find the y-intercept using the points in the table?

Tip: Remember, the x-intercept is where the line crosses the x-axis, and at this point, y=0y = 0.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
X-Intercept

Formulas

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

Theorems

Linear Equation

Suitable Grade Level

Grades 8-10