Calculus: Difference Quotient and Slope Concepts

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Questions and Answers

What are the steps to find the Difference Quotient?

find f(x+h), 2. simplify f(x+h) - f(x), 3. divide the result by h

What does the Slope Formula represent?

It represents the ratio of the change in y to the change in x between two points.

A Positive Slope rises from left to right.

True

A Negative Slope falls from left to right.

True Signup and view all the answers

When would you use the Point-Slope Form Equation?

When given two points and a slope, or two sets of ordered pairs. Signup and view all the answers

What does the Slope-Intercept Form Equation indicate?

y-intercept = b Signup and view all the answers

What is the equation for a General Form?

Ax + By = C, where A cannot be a fraction Signup and view all the answers

Parallel lines have the same slope.

True Signup and view all the answers

Perpendicular lines have the same slope.

False Signup and view all the answers

What is the definition for a Constant Function Graph?

A graph that has a horizontal line. Signup and view all the answers

What kind of function does F(x)= x represent?

Linear Function Signup and view all the answers

What does the Absolute Value Function Graph represent?

F(x) = |x| represents the graph of absolute values. Signup and view all the answers

What is the equation for vertical shifts?

y = f(x) + c or y = f(x) - c Signup and view all the answers

What is the equation for horizontal shifts?

y = f(x + c) or y = f(x - c) Signup and view all the answers

What does the equation y = -f(x) represent?

Reflection about the x-axis Signup and view all the answers

What does the equation y = f(-x) represent?

Reflection about the y-axis Signup and view all the answers

What does vertically stretching mean in terms of the equation?

y = cf(x), where c > 1 Signup and view all the answers

What does vertically shrinking mean in terms of the equation?

y = cf(x), where 0 < c < 1 Signup and view all the answers

What does horizontally stretching mean?

y = f(cx), where c > 1 Signup and view all the answers

What does horizontally shrinking mean?

y = f(cx), where 0 < c < 1 Signup and view all the answers

Study Notes

Difference Quotient

Represents the average rate of change of a function over an interval.
Calculated using f(x+h), simplifying f(x+h) - f(x), and then dividing by h.

Slope Formula

Determines the steepness of a line.
Expressed as (y₂ - y₁) / (x₂ - x₁).

Positive Slope

Indicates a line that rises from left to right.
Represents a direct relationship between variables.

Negative Slope

Indicates a line that falls from left to right.
Represents an inverse relationship between variables.

Point-Slope Form Equation

Used to write the equation of a line when given a point and slope.
Form: y - y₁ = m(x - x₁).

Slope-Intercept Form Equation

Standard form for expressing linear equations.
Form: y = mx + b, where b represents the y-intercept.

General Form Equation

Represents a linear equation in the format Ax + By = C.
A cannot be a fraction for standard representation.

Parallel Lines

Have the same slope.
Never intersect in a plane.

Perpendicular Lines

Have slopes that are opposite reciprocals.
Intersect at 90-degree angles.

Constant Function Graph

Horizontal line indicating the output is constant.
Form: y = k, where k is a constant.

Identity Function Graph

Straight line at a 45-degree angle.
Form: y = x, representing equality of input and output.

Absolute Value Function Graph

V-shaped graph reflecting distance from zero.
Form: y = |x|.

Standard Quadratic Function Graph

Parabola opening upwards.
Form: y = x².

Square Root Function Graph

Begins at the origin and increases gradually.
Form: y = √x.

Standard Cubic Function Graph

S-shaped curve that passes through the origin.
Form: y = x³.

Cube Root Function Graph

Similar to the cubic function but flattens out.
Form: y = ∛x.

Vertical Shifts

Result from adding or subtracting a constant to the function.
Form: y = f(x) + c (up) or y = f(x) - c (down).

Horizontal Shifts

Result from adding or subtracting a constant within the function's argument.
Form: y = f(x + c) (left) or y = f(x - c) (right).

Reflection about the x-axis

Inverts the graph upside down.
Form: y = -f(x).

Reflection about the y-axis

Flips the graph over the y-axis.
Form: y = f(-x).

Vertically Stretching

Increases the steepness of the graph.
Form: y = cf(x), where c > 1.

Vertically Shrinking

Decreases the steepness of the graph.
Form: y = cf(x), where 0 < c < 1.

Horizontally Stretching

Expands the graph away from the y-axis.
Form: y = f(cx), where c > 1.

Horizontally Shrinking

Compresses the graph towards the y-axis.
Form: y = f(cx), where 0 < c < 1.

Inverse of a Function

A function that reverses the effect of the original function.
Originates from swapping the x and y variables.

Horizontal Line Test for Inverse Functions

A method to determine if a function is invertible.
If any horizontal line intersects the graph more than once, it's not invertible.

Linear Function Properties

Form: F(x) = x, defined over all real numbers.
Domain: (-∞, ∞), Range: (-∞, ∞).

Quadratic Function Properties

Form: F(x) = x².
Domain: (-∞, ∞), Range: [0, ∞).

Cubic Function Properties

Form: F(x) = x³.
Domain: (-∞, ∞), Range: (-∞, ∞).

Rational Inverse Function Properties

Form: F(x) = 1/x.
Domain: (-∞, 0) U (0, ∞), Range: (-∞, 0) U (0, ∞).

Rational Squared Function Properties

Form: F(x) = 1/x².
Domain: (-∞, 0) U (0, ∞), Range: (0, ∞).

Absolute Value Function Properties

Form: F(x) = |x|.
Domain: (-∞, ∞), Range: [0, ∞).

Square Root Function Properties

Form: F(x) = √x.
Domain: [0, ∞), Range: [0, ∞).

Exponential Function Properties

Form: F(x) = e^x.
Domain: (-∞, ∞), Range: (0, ∞).

Natural Logarithm Function Properties

Form: F(x) = ln(x).
Domain: (0, ∞), Range: (-∞, ∞).

Sine Function Properties

Form: F(x) = sin(x).
Domain: (-∞, ∞), Range: [-1, 1].

Quadratic Formula

Used to find roots of quadratic equations.
Formula: x = (-b ± √D) / 2a, where D = b² - 4ac.

Compound Interest Formula

Formula for calculating interest on principal over time.
A = P(1 + r/n)^(nt), where A = amount, P = principal, r = rate, n = times compounded, t = time.

Simple Interest Formula

Calculates interest earned or paid on principal alone.
Formula: I = Prt, where I = interest, P = principal, r = rate, t = time.

Complex Numbers

Expressed in the form a + bi, where a is the real part and bi is the imaginary part.
Utilized in advanced mathematics and engineering.

Factor by Grouping

A method for factoring polynomials.
Groups terms in sets to factor common factors from each group.

Discriminant Formula

D = b² - 4ac, indicates the nature of roots for quadratic equations.
If D > 0: two distinct real solutions, if D = 0: one real solution (double root), if D < 0: no real solutions.

Horizontal Stretch

Results from manipulations that expand the graph horizontally.
Defined by adjustments to the function's input variable, typically applied by altering coefficients.

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Description

This quiz explores fundamental concepts in calculus, focusing on the Difference Quotient and the Slope Formula. You will learn how to determine positive and negative slopes, when to apply the Point-Slope Form, and the significance of the Slope-Intercept Form. Master these essential topics to enhance your understanding of calculus.