Applying Linear Functions and Algebraic Techniques
This page focuses on practical applications of linear functions and algebraic techniques, including calculating function values, determining parameters, and finding antecedents.
Calculating Function Values:
For the function P(x) = 3x + 2, we can calculate P(-1):
P(-1) = 3(-1) + 2 = -3 + 2 = -1
Determining Function Parameters:
Given P(x) = 5x + p and P(-2) = 3, we can find p:
5(-2) + p = 3
-10 + p = 3
p = 13
Therefore, P(x) = 5x + 13
Example: For the function P(x) = mx - 3, if P(4) = 12, we can determine m:
mx4 - 3 = 12
4m = 15
m = 15/4 = 3.75
Thus, P(x) = 3.75x - 3
Finding Antecedents:
To find the antecedent of -1 for f(x) = -1/5x + 2:
-1/5x + 2 = -1
-1/5x = -3
x = 15
Highlight: The antecedent is the input value (x) that produces a specific output value for a given function.
Determining a Linear Function from Two Points:
To find the function f(x) such that f(-6) = 5 and f(3) = -1:
m = (f(3) - f(-6)) / (3 - (-6)) = (-1 - 5) / 9 = -2/3
Using the point-slope form: f(x) = -2/3x + b
Substituting f(-6) = 5: 5 = -2/3(-6) + b
5 = 4 + b
b = 1
Therefore, f(x) = -2/3x + 1
Vocabulary: The slope formula used here is (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.
The page concludes with additional calculations and formulas related to linear functions and slopes, reinforcing the practical application of these factorisation formule and development techniques in solving algebraic problems.
