Gr12 Mathematics: Ch 5.6 Sketching Graphs

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What is the effect of a > 0 on the graph of y = ax^3 + bx^2 + cx + d?

The graph rises to the right and falls to the left. (correct)

The graph is a horizontal line.

The graph is a straight line.

The graph falls to the right and rises to the left.

How do you find the y-intercept of a cubic function f(x) = ax^3 + bx^2 + cx + d?

Solve f(x) = 0.

Set x = 0. (correct)

Solve f'(x) = 0.

Set x = 1.

What is the purpose of finding the first derivative f'(x) when sketching a cubic graph?

To find the local maximum and local minimum.

To find the x-intercepts.

To find the stationary points. (correct)

To find the y-intercepts.

What is a point of inflection in a cubic graph?

A point where the graph changes concavity. Signup and view all the answers

What is the condition for a graph to be concave up?

f''(x) > 0. Signup and view all the answers

How can you determine the shape of a cubic graph?

By considering the sign of a. Signup and view all the answers

What is a local minimum in a cubic graph?

A point where the function changes from decreasing to increasing. Signup and view all the answers

What is the first step in the general method for sketching cubic graphs?

Determine the shape. Signup and view all the answers

What happens to the graph of y = ax^3 + bx^2 + cx + d when a < 0?

The graph falls to the right and rises to the left Signup and view all the answers

What is the purpose of solving f(x) = 0 when finding the intercepts of a cubic function?

To find the x-intercepts Signup and view all the answers

What is the relation between the concavity of a curve and its gradient?

The gradient is increasing when the curve is concave up Signup and view all the answers

What is the condition for a point to be a local maximum in a cubic graph?

The function changes from increasing to decreasing Signup and view all the answers

What do you need to find to determine the x-coordinates of the stationary points of a cubic function?

The first derivative f'(x) Signup and view all the answers

What happens at a point of inflection in a cubic graph?

The graph changes concavity Signup and view all the answers

What is the purpose of finding the x-intercepts when sketching a cubic graph?

To find the points where the graph crosses the x-axis Signup and view all the answers

What is the relation between the sign of a and the orientation of a cubic graph?

a > 0 corresponds to a graph that rises to the right Signup and view all the answers

If a > 0, what is the relationship between the x-intercepts and the y-intercept of a cubic graph?

The x-intercepts may be to the left or right of the y-intercept Signup and view all the answers

What is the relation between the stationary points and the local maximum and local minimum?

A stationary point may be a local maximum or local minimum, or neither Signup and view all the answers

What is the purpose of finding the x-coordinates of the stationary points?

To classify the stationary points as local maximum or local minimum Signup and view all the answers

What is the relationship between the concavity of a curve and its gradient?

The concavity of a curve indicates whether the gradient is increasing or decreasing Signup and view all the answers

How can you determine the orientation of a cubic graph?

By considering the sign of a Signup and view all the answers

What is the condition for a graph to change concavity?

f''(x) = 0 Signup and view all the answers

What is the purpose of finding the second derivative f''(x) when sketching a cubic graph?

To determine the concavity of the graph Signup and view all the answers

How can you determine whether a stationary point is a local maximum or local minimum?

By considering the sign of f''(x) at the stationary point Signup and view all the answers