Math Problem Statement

What is the zero of g?

Solution

The graph shows the linear function gg passing through the points (8,3)(-8, 3) and (3,4.5)(-3, -4.5). To determine the zero of the function, we need to find the x-intercept, which is the value of xx when y=0y = 0.

Let's start by finding the equation of the line in slope-intercept form y=mx+by = mx + b.

Step 1: Calculate the slope mm

The formula for slope between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

Substituting the given points (8,3)(-8, 3) and (3,4.5)(-3, -4.5):

m=4.533(8)=7.55=1.5m = \frac{-4.5 - 3}{-3 - (-8)} = \frac{-7.5}{5} = -1.5

So, the slope m=1.5m = -1.5.

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

Now, using the point-slope form of the equation of a line, yy1=m(xx1)y - y_1 = m(x - x_1), we can substitute one of the points, say (8,3)(-8, 3), and the slope m=1.5m = -1.5:

y3=1.5(x(8))=1.5(x+8)y - 3 = -1.5(x - (-8)) = -1.5(x + 8)

Simplifying:

y3=1.5x12y - 3 = -1.5x - 12 y=1.5x9y = -1.5x - 9

Step 3: Find the zero of the function

To find the zero, set y=0y = 0 and solve for xx:

0=1.5x90 = -1.5x - 9

Add 9 to both sides:

1.5x=91.5x = -9

Now, divide by 1.5:

x=6x = -6

Thus, the zero of the function gg is x=6x = -6.


Do you need further clarification or more details?

Here are 5 related questions to explore further:

  1. How can we verify that the equation of the line is correct using both points?
  2. What is the significance of the slope in a linear function?
  3. How do we interpret the y-intercept of the function?
  4. Can you find the zero of a quadratic function using a similar method?
  5. What is the connection between the slope and the steepness of a line?

Tip: When calculating the slope, always double-check the signs of your coordinates to ensure accuracy!

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + b

Theorems

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Suitable Grade Level

Grades 8-10