The Forms of a Quadratic Function are:
(1.) Standard Form/General Form
Note that the standard form is written in descending order of x
$
y = ax^2 + bx + c \\[3ex]
OR \\[3ex]
f(x) = ax^2 + bx + c \\[3ex]
Vertex = \left[-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right] \\[5ex]
$
(2.) Vertex Form
$
y = a(x - h)^2 + k \\[3ex]
OR \\[3ex]
f(x) = a(x - h)^2 + k \\[3ex]
Vertex = (h, k) \\[3ex]
h = x-coordinate\;\;of\;\;the\;\;vertex \\[3ex]
k = y-coordinate\;\;of\;\;the\;\;vertex \\[3ex]
$
(3.) Extended Vertex Form
$
f(x) = a(x - h)^2 + k ...Vertex\;\;Form \\[3ex]
f(x) = a\left(x + \dfrac{b}{2a}\right)^2 + \dfrac{4ac - b^2}{4a}...Extended\;\;Vertex\;\;Form \\[5ex]
Vertex = (h, k) \\[3ex]
\implies \\[3ex]
-h = \dfrac{b}{2a} \\[5ex]
h = -\dfrac{b}{2a} \\[5ex]
k = \dfrac{4ac - b^2}{4a} \\[5ex]
\implies \\[3ex]
\boldsymbol{Vertex = (h, k) = \left(-\dfrac{b}{2a}, \dfrac{4ac - b^2}{4a}\right)} \\[5ex]
$
Attempt all questions.
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Let: |
p < 0 p = −1 |
0 ≤ p ≤ 100 p = 1 |
p > 100 p = 101 |
---|---|---|---|
$p$ | − | + | + |
$p - 100$ | − | − | + |
$p(p - 100)$ | + | − | + |
Let: |
p < 13 p = 0 |
13 ≤ p ≤ 87 p = 17 |
p > 87 p = 87 |
---|---|---|---|
$p - 13$ | − | + | + |
$p - 87$ | − | − | + |
$(p - 13)(p - 87)$ | + | − | + |
Let: |
p < 0 p = −1 |
0 ≤ p ≤ 50 p = 1 |
p > 50 p = 51 |
---|---|---|---|
$p$ | − | + | + |
$p - 50$ | − | − | + |
$p(p - 50)$ | + | − | + |
Let: |
p < 12 p = 0 |
12 ≤ p ≤ 38 p = 16 |
p > 38 p = 39 |
---|---|---|---|
$p - 12$ | − | + | + |
$p - 38$ | − | − | + |
$(p - 12)(p - 38)$ | + | − | + |