Two Forms of a Quadra.c Func.on
Vertex Form
y = a(x – h)2 + k
a makes the parabola narrower or broader
h moves the parabola right or left
k moves the parabola up or down
(h, k) are the coordinates of the vertex point
x = h is the axis of symmetry
Standard Form
y = ax 2 + bx + c
We're going to let a = 1 for this next section.
So we'll work with equations in the form
2
y = x + bx + c
The x-­‐intercepts (or “roots”) are where the parabola crosses through the x-­‐axis. Parabolas don’t always have x-­‐
intercepts. If they do, they can be found by subs.tu.ng zero for y in the equa.on and then solving for x.
x=
−b
This equa.on gives the x-­‐coordinate of the 2a
vertex point. It also gives the equa.on for the axis of symmetry. (So, –b/2a = h.)
axis of symmetry
Graphing from vertex form is preOy easy. There is a way to change standard form equa.ons to vertex form equa.ons. The method is know as “comple.ng the square.” The idea behind this is to change the trinomial into one that is a perfect square trinomial. In the example below, I have moved the added three off to the side and added in a four. I also subtracted four form the three in order to keep the equa.on equivalent. The, I factored the perfect square trinomial.
y = x 2 + 4x + 3
y = x 2 + 4x
+3
y = x + 4x + 4
+ 3– 4
2
y = (x + 2) – 1
2
This is now in vertex form. The x-­‐intercepts (roots) can be found by le.ng y=0...
y = (x + 2)2 – 1
0 = (x + 2)2 – 1
1 = (x + 2)2
1 = (x + 2)2
±1 = x + 2
x = –2 ± 1
So x = –3, –1
For 1–10, change each standard form equa.on into vertex form, then graph. Find the roots. Sketch the axis of symmetry.
1) y = x 2 + 4x + 3 2) y = x 2 – 6x – 5 3) y = x 2 + 10x + 18 4) y = x 2 – 8x + 20
5) y = x 2 – 2x – 8 6) y = x 2 – 8x + 15 7) y = x 2 – 7x + 10
For #7, you might no.ce that the trinomial factors to (x–2)(x–5) and so solving for the roots would have been a liOle easier to factor and then use the zero product property. On the next 3 you might see if you can find the roots this way.
8) y = x 2 – 7x + 12 9) y = x 2 – 5x + 6 10) y = x 2 + 5x – 14
Conver.ng Standard Form to Vertex Form for a≠1 When “a” is the GCF for the trinomial.
• Factor out “a”
• complete the square for the trinomial
• distribute a back in
Example...
y = 5x 2 − 10x − 15
y = 5[x 2 − 2x
−3 ]
y = 5[x − 2x + 1
2
− 3 −1 ]
y = 5[ (x − 1) − 4 ]
2
y = 5(x − 1)2 − 20
When “a” is not the GCF for the trinomial.
• Factor out “a”
• complete the square for the trinomial
• distribute a back in
Example...
y = 5x 2 − 4x − 1
4
1 ⎤
⎡
y = 5 ⎢ x2 − x
−
5
5 ⎥⎦
⎣
4
16
1
16 ⎤
⎡
y = 5 ⎢ x2 − x +
− −
5
100
5 100 ⎥⎦
⎣
2
⎡⎛
4⎞
36 ⎤
y = 5 ⎢⎜ x − ⎟ −
⎥
10 ⎠ 100 ⎦
⎣⎝
2
4⎞
9
⎛
y = 5⎜ x − ⎟ −
⎝
10 ⎠
5
Convert each standard form quadra.c func.on into vertex form...
1) y = 5x 2 − 10x − 15 2) y = 3x 2 + 12x − 96 3) y = 2x 2 + 12x − 14 5) y = 2x 2 − 24x + 72 (Graph this one.) 6) y = 2x 2 − 5x + 3 Convert each standard form quadra.c func.on into vertex form, and graph...
8) y = 5x 2 + 7x − 12 9) y = 4x 2 + 5x + 2 10) y = 3x 2 + 2x − 5 4) y = 4x 2 − 40x + 84
7) y = 3x 2 + 8x + 5
11) y = −2x 2 − 4x + 3
Practice ....
For each exercise, if it is in vertex form, put it into standard form. If it is in standard form, put it in vertex
form. Find the axis of symmetry. Find the roots. Graph the quadratic function and also the axis of
symmetry.
1) y= x2+4x–9
2) y= –x2+5x+6
3) y= –(x–2)2 +1
4) y= x2–2x–8
5) y= x2–4x+3
6) y= –x2+2x+3
7) y= –x2–6x+15
8) y= x2–14x+13
9) y= x2+2x+18
10) y+2 = x2–10x+25
11) y–5 = ⅓ (x+2)2
Quiz...Quadra.c Func.on -­‐-­‐ Two Forms & Roots Algebra 1 ______________________________
1. Give vertex form for the quadra.c func.on: ____________________________________
Give standard form for the quadra.c func.on: ____________________________________
For 2–4, change each standard form equa.on into vertex form, find the roots. Graph the func.on. Sketch the axis of symmetry and give its equa.on. Be sure to label the coordinates for the vertex of the parabola, and the roots, if they aren’t easily seen from the graph.
2. y = x 2 – 8x + 20 3. y = x 2 + 10x + 18 4. y = 4x 2 + 5x + 2 5. Could you find the roots for any of the func.ons in 2 – 4 by factoring? If so, which ones, and how do you know this?
Quiz...Quadra.c Func.on -­‐-­‐ Two Forms & Roots Algebra 1 ______________________________
April 16
1. Give vertex form for the quadra.c func.on: ____________________________________
Give standard form for the quadra.c func.on: ____________________________________
For 2–4, change each standard form equa.on into vertex form, find the roots. Graph the func.on. Sketch the axis of symmetry and give its equa.on. Be sure to label the coordinates for the vertex of the parabola, and the roots, if they aren’t easily seen from the graph.
2. y = x 2 – 8x + 20 3. y = x 2 + 10x + 18 4. y = 4x 2 + 5x + 2 5. Could you find the roots for any of the func.ons in 2 – 4 by factoring? If so, which ones, and how do you know this?