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.

Development and Factorization Formulas

This page presents essential formulas for développement (expansion) and factorisation (factoring) in algebra. These formulas are crucial for simplifying and manipulating algebraic expressions.

Development Formulas:

1. Distributive property: k$a + b$ = ka + kb and k$a - b$ = ka - kb
2. Product of binomials: a + b$$c + d = ac + ad + bc + bd
3. Square of a sum: $a + b$² = a² + 2ab + b²
4. Square of a difference: $a - b$² = a² - 2ab + b²
5. Difference of squares: a + b$$a - b = a² - b²

Factorization Formulas:

1. Common factor: ab + ac = a$b + c$
2. Difference of terms: ab - ac = a$b - c$
3. Perfect square trinomial (sum): a² + 2ab + b² = $a + b$²
4. Perfect square trinomial (difference): a² - 2ab + b² = $a - b$²
5. Difference of squares: a² - b² = a + b$$a - b

Highlight: Memorizing these formulas will greatly enhance your ability to manipulate algebraic expressions efficiently.

The page also includes a graph illustrating a linear function, which is described by the general form f$x$ = mx + p, where m is the slope and p is the y-intercept.

Example: For the function f$x$ = x - 2, the slope $m$ is 1, and the y-intercept $p$ is -2.

Another function, g$x$ = 2x + 3, is also mentioned, demonstrating how different values of m and p affect the graph of a linear function.

Vocabulary: In the context of linear functions, 'm' represents the slope, which indicates the steepness and direction of the line, while 'p' represents the y-intercept, the point where the line crosses the y-axis.