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

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

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

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

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

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

It implies that the function is linear.

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

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

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

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

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

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

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

It indicates that the rate of change is zero.

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

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

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

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

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

1. Calculate f(2) = 8. 2. Calculate f(0) = 0. 3. Apply the formula: (f(2) - f(0)) / (2 - 0) = (8 - 0) / 2 = 4.

1. Identify the y-values corresponding to the given x-values. 2. Subtract the y-values. 3. Subtract the x-values. 4. Divide the change in y by the change in x.

1. Find the second derivative of the function. 2. If the second derivative is positive, it's concave up. 3. If the second derivative is negative, it's concave down.

1. Find the vertex of the parabola. 2. If the coefficient of $x^2$ is positive, the function is increasing to the right of the vertex. 3. If the coefficient of $x^2$ is negative, the function is increasing to the left of the vertex.

1. Calculate the rate of change between consecutive points. 2. If the rate of change is constant, the function is linear.

1. Calculate $f(4) = 3(4) - 2 = 10$. 2. Calculate $f(-1) = 3(-1) - 2 = -5$. 3. Apply the formula: $\frac{f(4) - f(-1)}{4 - (-1)} = \frac{10 - (-5)}{5} = 3$.

1. Find the derivative: $f'(x) = 2x - 4$. 2. Set the derivative equal to zero: $2x - 4 = 0$. 3. Solve for x: $x = 2$.

1. Calculate $f(2) = -(2)^2 + 5(2) - 3 = 3$. 2. Calculate $f(0) = -(0)^2 + 5(0) - 3 = -3$. 3. Apply the formula: $\frac{f(2) - f(0)}{2 - 0} = \frac{3 - (-3)}{2} = 3$.

1. Find the second derivative: $f'(x) = 4x - 3$, $f''(x) = 4$. 2. Since $f''(x) = 4 > 0$, the function is concave up for all x.

1. Find the vertex: $x = \frac{-b}{2a} = \frac{-6}{2(-1)} = 3$. 2. Since the coefficient of $x^2$ is negative, the function is decreasing for $x > 3$.