The idea of this lecture is be able to sketch the graph of a quadratic equations like: y= x2 - 4x + 5
I. graphing a (x - h )2 + k (7.5 p.617)
1) Let's start with y = x2.
GRaph the equation. Start by building a table of x,y values.
| x | y |
| 0 | |
| +/-1 | |
| +/- 2 | |
| +/- 3 |
You see that the graph opens upward. The graph is called a parabola or quadratic curve. The curve has a minimum (lowest point on the curve) . It is called a vertex. If the curve opens downward, the vertex is a maximum.
Also, Notice the symmetry. The y-axis is the axis of symmetry or line of symmetry.
2) Let's make make some change to the equation and see what changes occur in the graph.
graph : y = x2 - 3
You can build a table of x,y values.
| x | y |
| 0 | |
| +/-1 | |
| +/- 2 | |
| +/- 3 |
Compare with the previous table. Notice how all the y-values have been shifted 3 down. All the vertical values (y values) are moved down by 3. Graph the parabola above. (use a pencil). You should see that the parabola has been moved by 3 down.
if f(x) = x2 and g(x) = x2 - 3 you have g(x) = f(x) - 3
3) What if you add a number before the squaring ?
y = (x - 2 )2
Try to graph the new parabola/
The new parabola is shifted, unexpectively, to the right.
4) try now y = (x+3)2
5) try 2 transformations together graph: y= (x+3)2 - 2
What is the vertex ?
THESE KIND OF TRANSFORMATIONS ARE SAID TO BE RIGID.
the parabola y=x2 moves up/down right/left but the shape stays the same.
Other transformations can make the parabola looks fatter or thinner
or make the parabola opens downward instead of upward. It won't affect the vertex.
6) try y = 2x2
The coefficient 2 is greater than 1. Compared to y = x2, you are doubling the vertical distances.
Trace the graph. It should look thinner. The graph rises faster.
7) Try now y = 1/3 x2. Now the vertical distances (the y-values) are one-third the vertical
distances of y = x2. The parabola should get fatter. graph it.
8) what if you have a negative coefficient ? like y = - x2 . The vertical distances become negative when compared to y=x2.
The parabola opens downward.
9) Try now to combine 3 kinds to transformation.
A) sketch the parabola B) find the vertex C) find the x-intercepts (that is solve y=0) D) find the line of symmetry
y = 2 (x+3)2 - 2
10) t(x) = 1/2 (x-3)2 + 5 (from book p. 622)
A) sketch the parabola B) find the vertex C) find the x-intercepts (y=0) D) Find the line of symmetry
11) f(x) = -2 (x+3)2 + 5 (from book p. 622)
A) sketch the parabola B) find the vertex C) find the x-intercepts (y=0) D) Find the line of symmetry
12) f(x) = (x-3)2 - 5 (from book p. 621)
A) sketch the parabola B) find the vertex C) find the x-intercepts (y=0) D) Find the line of symmetry
II. graphing ax2 + bx + c (7.6 p.628)
1) y = x2 - 6x +4/ (from book p.630
A) sketch the parabola B) find the vertex C) find the x-intercepts (y=0) D) Find the line of symmetry
TO answers these questions, the best way is to complete the square first.
2) Try f(x)= x2 - 4x + 7
3) y = x2 - 6x + 5
4) y = -x2 + 4x +1
PART II - QUIZ
from assignments due today
PART III assignments for next time.
p.624 2, 4, 6, 12,
p.626 19, 20, 21, 22, 24, 26
p.634 1,2,3, 4
p. 636 13, 15, 16, 20
p. 407 17, 14, 15, 18, 19, 27, 25, 40, 43, 50 , 51, 52. 53, 58
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