Question

vertex form : f(x) = a(x - h)^2 + k
remember... like absolute value functions → + a opens up, - a opens down
info. you must figure out →

1. vertex coordinate ( h , k ) ...\h\ comes out of the ( ) opposite, \k\ is exactly what’s given.
2. axis of symmetry (dotted line running up & down, center of the parabola) x = h

example : f(x) = - 2(x + 3)^2 + 4 **notice that \a\ is negative, so the parabola should open down

1. identify vertex → ( - 3 , 4 )
2. axis of symmetry → x = h , so, x = - 3 (dotted line down the center)

determine the vertex and equation for the axis of symmetry for each quadratic function below. also state the direction of the parabola’s opening.

1. y=(x + 4)^2 + 3 vertex: a.o.s.: direction of opening:
2. y=-2(x - 1)^2 + 7 vertex: a.o.s.: direction of opening:
3. y = 1/2(x + 2)^2 - 1 vertex: a.o.s.: direction of opening:
4. f(x)=(x - 3)^2 vertex: a.o.s.: direction of opening:
5. y = 3x^2 - 2 vertex: a.o.s.: direction of opening:

Explanation:

Step1: Recall vertex - form rules

For a quadratic function in vertex - form $y = a(x - h)^2+k$, the vertex is $(h,k)$ and the axis of symmetry is $x = h$. If $a>0$, the parabola opens up; if $a < 0$, the parabola opens down.

Step2: Analyze $y=(x + 4)^2+3$

Here $a = 1$, $h=-4$, $k = 3$.
Vertex: $(-4,3)$
Axis of symmetry (a.o.s.): $x=-4$
Since $a = 1>0$, direction of opening: up

Step3: Analyze $y=-2(x - 1)^2+7$

Here $a=-2$, $h = 1$, $k = 7$.
Vertex: $(1,7)$
Axis of symmetry: $x = 1$
Since $a=-2<0$, direction of opening: down

Step4: Analyze $y=\frac{1}{2}(x + 2)^2-1$

Here $a=\frac{1}{2}$, $h=-2$, $k=-1$.
Vertex: $(-2,-1)$
Axis of symmetry: $x=-2$
Since $a=\frac{1}{2}>0$, direction of opening: up

Step5: Analyze $f(x)=(x - 3)^2=(x - 3)^2+0$

Here $a = 1$, $h = 3$, $k = 0$.
Vertex: $(3,0)$
Axis of symmetry: $x = 3$
Since $a = 1>0$, direction of opening: up

Step6: Analyze $y = 3x^2-2=3(x - 0)^2-2$

Here $a = 3$, $h = 0$, $k=-2$.
Vertex: $(0,-2)$
Axis of symmetry: $x = 0$
Since $a = 3>0$, direction of opening: up

Answer:

1. vertex: $(-4,3)$; a.o.s.: $x=-4$; direction of opening: up
2. vertex: $(1,7)$; a.o.s.: $x = 1$; direction of opening: down
3. vertex: $(-2,-1)$; a.o.s.: $x=-2$; direction of opening: up
4. vertex: $(3,0)$; a.o.s.: $x = 3$; direction of opening: up
5. vertex: $(0,-2)$; a.o.s.: $x = 0$; direction of opening: up