Download presentation
Presentation is loading. Please wait.
Published byMae Cooper Modified over 9 years ago
1
MATH 31 LESSONS PreCalculus 8. Sketching Functions
2
A. Relations and Functions Whenever one variable (y) is affected by another variable (x), we say they form a relation. A function is a special relation in which every x-value has at most one y-value. That is, if an x-value could have more than one y-value, it is not a function. All functions are relations, but not all relations are functions.
3
Vertical Line Test If a relation is a function, then any vertical line can cross the graph at most once. If the vertical line crosses the graph more than once, then it is not a function.
4
e.g. Which of the following relations are functions?
5
This is a function. The vertical lines never cross twice (or more).
6
This is not a function. A vertical line can cross twice.
7
This is not a function. A vertical line can cross twice.
8
This is a function. The vertical lines never cross twice (or more).
9
B. Parabolas A parabola is a function of the form
10
Basic form The basic (simplest) form of a parabola is
11
Vertex: (0, 0) Opens upward y x
12
Standard form The standard form (also called the completed-square form) of a parabola is or
13
Vertex: (h, k) y x k h V(h, k)
14
Vertex: (h, k) Axis of symmetry: x = h y x x = h V(h, k) The axis of symmetry cuts the graph “in half”
15
Vertex: (h, k) Axis of symmetry: x = h Max / Min Value: y = k y x y = k V(h, k) If the graph opens down, k is a maximum value. If the graph opens up, k is a minimum value.
16
If a < 0, then the parabola opens downward e.g.y = - x 2 y = -3 (x - 2) 2 + 7 y x
17
If a > 0, then the parabola opens upward e.g.y = x 2 + 11 y = 3 (x - 1) 2 - 4 y x
18
y x y = k x = h V(h, k)
19
Ex. 1Find the vertex of by completing the square. Try this example on your own first. Then, check out the solution.
20
Isolate the x-variables
21
Factor out the coefficient of the squared term
22
Determine the constant that is needed to make a perfect square, by squaring half of the middle (linear) term.
23
Add the coefficient to both sides. Don’t forget that the 16 is multiplied by 3 on the right side. You must do that on the left as well.
24
Complete the square
25
Put the parabola in standard form
26
Vertex: (-4, -28)
27
Ex. 2Fully sketch Identify key features of the graph, including vertex, max/min, axis of symmetry, and intercepts. Try this example on your own first. Then, check out the solution.
28
Complete the square to put in standard form
29
Isolate the x-variables
30
Factor out the coefficient of the squared variable
31
Find the constant required to make a perfect square:
32
Add the constant to both sides. Don’t forget that the 25 is actually negative, due to the coefficient in front of the brackets.
33
Complete the square
34
Vertex: (5, 4) Opens downward (a < 0) Axis of symmetry: x = 5 Max Value: y = 4 Put in standard form.
35
Find intercepts y-int: (x = 0)
36
Find intercepts y-int: (x = 0) (0, -21)
37
x-int: (y = 0)
38
x-int: (y = 0)
39
x-int: (y = 0)
40
x-int: (y = 0) (3, 0) (7, 0)
41
Sketch y x y = 4 x = 5 V(5, 4) 37
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.