Quiz: Understanding Affine Functions and Inequalities — 5 domande

1. How does an affine function differ from a linear function in terms of their graphs and algebraic expressions?

A. An affine function has a variable slope, while a linear function has a fixed slope of 1.

B. An affine function always passes through the origin, but a linear function has a y-intercept p ≠ 0.

C. An affine function has a nonzero y-intercept and is written as f(x) = mx + p, while a linear function passes through the origin and is written as f(x) = mx.

D. An affine function is a curve, whereas a linear function is always a straight line.

Answer: (C)

Spiegazione

An affine function is expressed as f(x) = mx + p with p representing the y-intercept, so its graph may not pass through the origin. A linear function is a special case of an affine function where p = 0, so its graph passes through the origin and is written as f(x) = mx. Review: Definition and properties of affine functions. Course evidence: "- An affine function is defined on the entire real line, ] -∞ ; +∞ [, and can be written as f(x) = mx + p. - The number p is called the ordonnée à l'origine, representing the y-intercept of the graph. - The number m is called the coefficient directeur,…"

2. What is an affine function?

A. A function that always passes through the origin

B. A function that can be expressed as f(x) = mx + p, where m and p are real numbers

C. A function that is only defined on positive real numbers

D. A function with a quadratic form

Answer: (B)

Spiegazione

An affine function is defined as a function that can be expressed in the form f(x) = mx + p, where m and p are real numbers, representing the slope and y-intercept respectively. Review: Definition and properties of affine functions. Course evidence: "A function defined on the interval ] -∞ ; +∞ [ that can be expressed in the form f(x) = mx + p, where m and p are real numbers."

3. How do the methods for determining the coefficient directeur and the constant term differ when finding the affine function expression from two points?

A. The coefficient directeur is found by calculating the ratio of vertical to horizontal changes between the two points, while the constant term is found by substituting one point into the function and solving for p

B. Both the coefficient directeur and constant term are directly given by the coordinates of the first point only

C. The coefficient directeur is found by substituting one point into the function, while the constant term is calculated from the ratio of vertical to horizontal changes

D. Both the coefficient directeur and constant term are found by averaging the x and y coordinates of the two points

Answer: (A)

Spiegazione

The coefficient directeur m is calculated as the ratio of the difference in y-coordinates to the difference in x-coordinates between two points, while the constant term p is found by substituting one point's coordinates into the function f(x) = mx + p and solving for p. Review: Determining affine function expression from two points. Course evidence: "- The coefficient directeur, denoted as m, can be calculated using the coordinates of two points A(xA, yA) and B(xB, yB) with the formula m = (yB - yA) / (xB - xA). This ratio measures the vertical change divided by the horizontal change between the points.…"

4. What is the purpose of calculating the coefficient directeur in determining an affine function from two points?

A. To calculate the average of the y-values

B. To determine the x-coordinate of the midpoint

C. To find the y-intercept of the affine function

D. To measure the vertical change divided by the horizontal change between the points

Answer: (D)

Spiegazione

The coefficient directeur is calculated as the ratio of the change in y to the change in x, which measures the vertical change divided by the horizontal change between the points. Review: Determining affine function expression from two points. Course evidence: "The coefficient directeur, denoted as m, can be calculated using the coordinates of two points A(xA, yA) and B(xB, yB) with the formula m = (yB - yA) / (xB - xA). This ratio measures the vertical change divided by the horizontal change between the points."

5. How do the monotonicity properties of affine functions differ based on the sign of their slope?

A. A positive slope makes the function strictly increasing, while a negative slope makes it strictly decreasing.

B. The slope sign does not affect the monotonicity of the affine function.

C. A positive slope makes the function constant, while a negative slope makes it decreasing.

D. A positive slope makes the function decreasing, while a negative slope makes it increasing.

Answer: (A)

Spiegazione

The source states that if the slope m > 0, the affine function is strictly increasing, indicating that positive slopes correspond to increasing functions, while negative slopes correspond to decreasing functions. Review: Monotonicity and variation of affine functions based on slope. Course evidence: "- If the slope m > 0, the affine function is strictly increasing on ] -∞ ; +∞ [."