Explain the difference between average rate of change in linear and quadratic functions.

Linear functions have a constant average rate of change, while quadratic functions have a changing average rate of change.

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Explain the difference between average rate of change in linear and quadratic functions.

Linear functions have a constant average rate of change, while quadratic functions have a changing average rate of change.

How is the average rate of change related to the slope of a secant line?

The average rate of change is equal to the slope of the secant line connecting two points on the function's graph.

What does concavity tell you about the rate of change of a function?

Concavity indicates whether the rate of change is increasing (concave up) or decreasing (concave down).

How does the sign of the average rate of change relate to whether a function is increasing or decreasing?

A positive average rate of change indicates the function is increasing, while a negative average rate of change indicates the function is decreasing.

Explain how to determine if a quadratic function is accelerating or decelerating.

If the average rate of change is increasing, the function is accelerating. If it's decreasing, the function is decelerating.

What is the significance of the vertex of a quadratic function in terms of rate of change?

The vertex represents the point where the rate of change changes direction (from decreasing to increasing or vice versa).

What does a constant average rate of change imply about the function?

It implies that the function is linear.

How does concavity relate to the second derivative of a function?

If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

How do you determine the concavity of a quadratic function?

If the coefficient of the $x^2$ term is positive, the function is concave up. If it is negative, the function is concave down.

What does the average rate of change approaching zero signify?

It signifies that the function's values are not changing significantly over the given interval, indicating a horizontal segment or a turning point.

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.