3.
Turning Points of Quadratic Graphs
Worksheet 5
A
B
A
Find the coordinates of A and B.
Find the coordinates of the turning point of f(x) using VERTEX FORM and CALCULUS.
USING CALCULUS: the slope of the
tangent at the turning point is zero.
Find the slopes of the tangents at A and B, respectively.
B
A
SLOPE OF TANGENT AT A.
Hence find the angle between the two tangents.
Find the average value of f(x) on the interval from 0 to 5.
SLOPE OF TANGENT AT B.
Worksheet 6
The graph of the function
Solve
is shown.
=0
Find the coordinates of the turning point using VERTEX FORM and CALCULUS.
USING CALCULUS: the slope of the
tangent at the turning point is zero.
Find the average value of g(x) on the interval from –3 to 1.
Find the coordinates of the point where the curve crosses the y-axis.
Find the equation of the tangent to at the point where the curve crosses the y-axis.
Find the acute angle to the horizontal that the tangent makes with the x-axis. Give your answer to the
nearest degree. Use a protractor to draw the tangent on the diagram above..
Worksheet 7
Find the coordinates of the turning point.
Use this turning point and your answers to part (i)
above to sketch the curve. Show your scale on
both axes.
SEC 2016 Paper 1 Question 7 (b)
Worksheet 8
USING VERTEX FORM
USING CALCULUS
4.
Quadratic and Cubic Equations and their Roots
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
13. Forming Polynomials
The shape of a polynomial is determined by the degree of the polynomial and its leading coefficient:
If the leading coefficient is positive, the curve goes up on the right
If the leading coefficient is negative, the arm goes down on the left
The way it intersects the x-axis is determined by the multiplicity of the root:
If the multiplicity of the root is odd, then the curve crosses the x-axis; the higher the
multiplicity the flatter the curve at the x-axis
If the multiplicity of the root is even, then the curve touches the x-axis; the higher the
multiplicity the flatter the curve at the x-axis
Example 1
Find an expression for this cubic polynomial.
The graph crosses the x-axis at x = -3, x = -1 and x = 2.
Thus
(x + 3) (x + 1) and (x - 2) are factors.
f(x) = k(x + 3) (x + 1) (x - 2)
NOW FIND THE VALUE OF K!!
(0, -6) is a point on the graph.
f(0) = k(0 + 3) (0 + 1) (0 - 2) = -6
-6k = -6
k=1
f(x) = 1(x + 3) (x + 1) (x - 2)
Multiplying out, we get:
14. Turning Points of Polynomials
Worksheet 29
(a)
Find the coordinates of the local maximum turning point and the local minimum turning point of f.
(b)
Find the coordinates of the point of inflection.
f(x)
f’(x
f’’(x)
MORE FOR YOU TO DO:
(c)
Find, in terms of x and y, the image of the point (x, y) under symmetry in the point of inflection.
(d)
Find the coordinates of the point where the graph of f cuts the y-axis.
(e)
Find the coordinates of the point where the tangent to the curve at this point crosses the x-axis.
(f)
Find the area enclosed between the curve and the x-axis.
15. Finding Areas by Integration
Example 1
NOW FIND THE COORDS OF THE TURNING POINTS AND THE POINT OF INFLECTION.
Example 2
(a)
SEC 2012 Paper 1 Question 5
(b)
Example 3
NOW FIND THE SHADED AREA BY SUBTRACTING THE FUNCTIONS!!
Worksheet 31
Find the equations of the tangents at the points of intersection between the line and the curve.
Find the angle between these tangents.