Consider the function y = 4 x - 3.
Find the gradient function of y = 4 x - 3.
Hence, sketch the graph of the gradient function.
For each of the following functions and their graphs:
Find a, the x-coordinate of the x-intercept of the gradient function.
For x<a, are the values of the gradient function above or below the x-axis?
For x > a, are the values of the gradient function above or below the x-axis?
y = \left(x - 5\right)^{2} - 3
y = x^{2} - 10 x + 29
y = - x^{2} - 10 x - 28
y = x^{2} + 10 x + 21
y = - \left(x + 7\right)^{2} + 5
y = - x^{2} + 12 x - 33
For each of the quadratic functions graphed below:
State the x-coordinate(s) of the x-intercept of the gradient function.
State the region(s) of the domain where the values of the gradient function are positive.
State the region(s) of the domain where the values of the gradient function are negative.
Hence, sketch the graph of the gradient function.
For each of the cubic functions graphed below:
State the x-coordinate(s) of the x-intercept of the gradient function.
State the region(s) of the domain where the values of the gradient function are positive.
State the region(s) of the domain where the values of the gradient function are negative.
Hence, sketch the graph of the gradient function.
Consider the functions f \left( x \right) = x^{5} and g \left( x \right) = x^{4}.
Sketch the graph of f \left( x \right) and its derivative.
Sketch the graph of g \left( x \right) and its derivative.
Determine whether the following statements are true or false:
The graph of the derivative can be found by translating and/or stretching the original function.
Near the origin, the derivative has a greater value than the function.
A function and its derivative have the same sign for all values of x.
If the degree of a function is even, then the degree of its derivative is odd and vice versa.
The graph of y = x^{5} is shown below labelled as A. Fiona then graphs the derivative of the function, labelling it as B. She repeats this with graph B to get graph C, then again to get graph D and again to get graph E.
Determine whether the following statements are true about this sequence of derivatives:
The derivative of a function is always positive when the function is negative, and negative when the function is positive.
Each graph is a function of the form a x^{n}.
For any value of x, the value of the derivative will always be greater than the value of the function.
The degree of the derivative is always different to the degree of the function.
If a function has degree n, what is the degree of the derivative?
Sketch the graph of the gradient function for each of the following graphs:
The diagram shows the graph of y = f \left( x \right):
State the interval(s) where the values of the derivative f' \left( x \right) are negative.
What happens to f' \left( x \right) for large values of x?
Draw the graph of y = f' \left( x \right).
The graph shows y=f(x) and y = g(x), where g(x)=f(x)+3:
Sketch the gradient function of y=f(x).
Sketch the gradient function of y=g(x).
What do you notice about the gradient functions of y=f(x) and y=g(x)? Explain why this is.
Consider the graph of y=f(x) shown:
Sketch a graph of a function that will have the same gradient function as y=f(x).
Consider the graph of f(x)=4x^2-2x+1 shown:
State whether the functions below will have the same gradient function as y=f(x):
The graph below shows the education level of students over the past 80 years. Sketch the graph of the gradient function.
The electrical resistance, R, of a component at temperature, t, is given by a quadratic function. The curve of R is shown on the graph.
Sketch the graph of \dfrac{d R}{d t}, the instantaneous rate of increase of resistance with respect to temperature.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 6 t^{2}
Sketch the graph of x(t) for 0 \leq t \leq 5.
Sketch the graph of v(t)=x'(t) for 0 \leq t \leq 5.
Use your graph to approximate the velocity of the object at t=3.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{3} - 4 t^{2}
Sketch the graph of x(t) for 0 \leq t \leq 4.
Sketch the graph of v(t)=x'(t) for 0 \leq t \leq 4.
Use your graph to approximate the velocity of the object at t=2.
The position (in metres) of an object along a straight line after t seconds is modelled by: x \left( t \right) = 3 t^{2} + 5 t + 2
Sketch the graph of x(t) for 0 \leq t \leq 4.
Sketch the graph of v(t)=x'(t) for 0 \leq t \leq 4.
Use your graph to approximate the velocity of the object at t=3.
The velocity (in metres per second) of a body moving in rectilinear motion after t seconds is modelled by: v \left( t \right) = t^{2} - 11 t + 24
Sketch the graph of v(t) for 0 \leq t \leq 4.
Sketch the graph of the acceleration function, a(t)=v'(t), for 0 \leq t \leq 4.
Use your graph to approximate the acceleration of the object at t=3.
A particle moves in a straight line. Its velocity (in metres per second), t seconds after passing the origin is given by: v = 2 t^{2} - 10 t
Sketch the graph of the acceleration function, a(t)=v'(t), for 0 \leq t \leq 4.
Use your graph to approximate the time when the acceleration is zero.