Math Problem Statement

Find a possible formula for the linear function h(x) if h(-30) = -80 and h(40) = 60. Write the equation in slope-intercept form.

Solution

To find the formula for the linear function h(x)h(x) given that h(30)=80h(-30) = -80 and h(40)=60h(40) = 60, we can use the information provided to determine the slope and then find the equation in slope-intercept form.

Step 1: Find the slope mm

The slope mm of a line through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is calculated as: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Using the points (30,80)(-30, -80) and (40,60)(40, 60):

m=60(80)40(30)=60+8040+30=14070=2m = \frac{60 - (-80)}{40 - (-30)} = \frac{60 + 80}{40 + 30} = \frac{140}{70} = 2

Step 2: Use the slope and one point to find the y-intercept bb

The slope-intercept form of the equation is: h(x)=mx+bh(x) = mx + b Using the slope m=2m = 2 and the point (30,80)(-30, -80):

80=2(30)+b-80 = 2(-30) + b 80=60+b-80 = -60 + b b=80+60=20b = -80 + 60 = -20

Final equation:

The equation of the linear function is: h(x)=2x20h(x) = 2x - 20

Would you like any further details or explanations on this?


Here are 5 related questions for further practice:

  1. What is the slope of the line that passes through the points (2,3)(2, 3) and (4,7)(4, 7)?
  2. How do you interpret the slope of a line in a real-world context?
  3. Can you find the equation of a line parallel to h(x)=2x20h(x) = 2x - 20 that passes through the point (0,5)(0, 5)?
  4. What is the y-intercept of the line h(x)=2x20h(x) = 2x - 20?
  5. How would the graph of h(x)=2x20h(x) = 2x - 20 change if the slope were negative?

Tip: Always double-check your calculations, especially when finding the slope, to ensure the accuracy of your linear equations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Functions

Formulas

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

Theorems

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Suitable Grade Level

Grades 8-10