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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.
How do you find the average rate of change of $f(x) = x^2 + 2x$ over the interval [0, 2]?
1. Calculate f(2) = 8. 2. Calculate f(0) = 0. 3. Apply the formula: (f(2) - f(0)) / (2 - 0) = (8 - 0) / 2 = 4.
Given a table of values, how do you estimate the average rate of change between two points?
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.
How to determine if a function is concave up or down given its equation?
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.
How do you find the interval where a quadratic function is increasing?
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.
How do you determine if a function is linear from a table of values?
1. Calculate the rate of change between consecutive points. 2. If the rate of change is constant, the function is linear.
How do you find the average rate of change of $f(x) = 3x - 2$ over the interval [-1, 4]?
1. Calculate $f(4) = 3(4) - 2 = 10$. 2. Calculate $f(-1) = 3(-1) - 2 = -5$. 3. Apply the formula: $\frac{f(4) - f(-1)}{4 - (-1)} = \frac{10 - (-5)}{5} = 3$.
How do you determine the x-value at which the instantaneous rate of change of $f(x) = x^2 - 4x + 5$ is equal to zero?
1. Find the derivative: $f'(x) = 2x - 4$. 2. Set the derivative equal to zero: $2x - 4 = 0$. 3. Solve for x: $x = 2$.
How do you find the average rate of change of $f(x) = -x^2 + 5x - 3$ over the interval [0, 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: $\frac{f(2) - f(0)}{2 - 0} = \frac{3 - (-3)}{2} = 3$.
How do you determine the concavity of $f(x) = 2x^2 - 3x + 1$?
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.
How do you find the interval where $f(x) = -x^2 + 6x - 8$ is decreasing?
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 $x > 3$.
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