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What is the formula for average rate of change?

$\frac{f(b) - f(a)}{b - a}$

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What is the formula for average rate of change?

$\frac{f(b) - f(a)}{b - a}$

How do you calculate the slope of a secant line between points (a, f(a)) and (b, f(b))?

$\frac{f(b) - f(a)}{b - a}$

Given a quadratic function in the form $f(x) = ax^2 + bx + c$ , how do you find the x-coordinate of the vertex?

$x = \frac{-b}{2a}$

What is the formula for the average rate of change of $f(x)$ over the interval $[x_1, x_2]$ ?

$\frac{\Delta y}{\Delta x} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

If $f(x) = ax + b$ , what is the average rate of change over any interval?

$a$

If $f(x) = ax^2 + bx + c$ , what is the average rate of change over the interval $[x_1, x_2]$ ?

$a(x_1 + x_2) + b$

What formula represents the slope of the secant line of a function $f(x)$ ?

$m_{sec} = \frac{f(x+h)-f(x)}{h}$

How is the average rate of change related to the difference quotient?

Average rate of change is equivalent to the difference quotient: $\frac{f(x_2)-f(x_1)}{x_2-x_1}$

How do you find the instantaneous rate of change?

$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

What is the general form for a linear equation?

$y = mx + b$

How can you identify intervals where a function is increasing or decreasing from its graph?

If the graph goes up from left to right, the function is increasing. If it goes down, it's decreasing.

How can you identify concavity (up or down) from a graph?

Concave up looks like a smile, concave down looks like a frown.

What does a steeper slope on a graph indicate about the rate of change?

A steeper slope indicates a larger rate of change (either increasing or decreasing more rapidly).

How can you approximate the average rate of change from a graph?

Draw a secant line between the two points and find its slope.

What does a horizontal line segment on a graph indicate about the rate of change?

It indicates that the rate of change is zero.

How does the graph of a quadratic function relate to its average rate of change?

The steepness of the curve indicates the magnitude of the average rate of change; the direction indicates whether it's increasing or decreasing.

How does the graph of $f(x) = x^2$ relate to its rate of change?

The graph is a parabola opening upwards. The rate of change is negative for $x < 0$ , zero at $x = 0$ , and positive for $x > 0$ .

How does the graph of $f(x) = -x^2$ relate to its rate of change?

The graph is a parabola opening downwards. The rate of change is positive for $x < 0$ , zero at $x = 0$ , and negative for $x > 0$ .

How can you identify the vertex of a quadratic function from its graph?

The vertex is the point where the graph changes direction (minimum or maximum point).

How can you identify the concavity from a graph?

A graph that opens upwards is concave up, and a graph that opens downwards is concave down.

What are the differences between the average rate of change and instantaneous rate of change?

Average rate of change: over an interval | Instantaneous rate of change: at a single point.

What are the differences between linear and quadratic functions in terms of their rates of change?

Linear: constant rate of change | Quadratic: changing rate of change.

What are the differences between concave up and concave down?

Concave Up: Increasing rate of change | Concave Down: Decreasing rate of change

Compare the average rate of change of $f(x) = 2x$ and $g(x) = x^2$ over the interval [1, 3].

f(x): Constant rate of change = 2 | g(x): Changing rate of change, average rate of change = 4

What is the difference between a secant line and a tangent line?

Secant line: intersects the curve at two points | Tangent line: touches the curve at one point.

Compare the concavity of $f(x) = x^2$ and $g(x) = -x^2$ .

f(x): Concave up | g(x): Concave down

Compare the average rate of change of a linear function with a positive slope and a linear function with a negative slope.

Positive slope: Average rate of change is positive | Negative slope: Average rate of change is negative

Compare the average rate of change of $f(x) = x$ and $g(x) = x^2$ as x approaches infinity.

f(x): Rate of change remains constant | g(x): Rate of change increases without bound

Compare the concavity of $f(x) = x^3$ and $g(x) = x^2$ around x = 0.

f(x): Changes concavity at x=0 (inflection point) | g(x): Always concave up

Compare the average rate of change of $f(x) = x$ and $g(x) = x+5$ .

f(x): Average rate of change is 1 | g(x): Average rate of change is 1