Here are a few recommended readings before getting started with this lesson.
All three graphs of the functions presented earlier have a very distinctive form. These curves have a specific name, which now will be properly introduced.
A parabola can be vertical or horizontal. A vertical parabola can open upward or downward. In comparison, a horizontal parabola can open to the left or the right.
Equations of parabolas always contain a variable raised to the second power. This is why the functions that represent vertical parabolas are called quadratic functions.
Name  Equation  Characteristics 

Standard Form  $y=ax_{2}+bx+c$  $a,$ $b,$ and $c$ are real numbers, $a =0,$ and $c$ is the $y$intercept of the parabola. 
Vertex Form  $y=a(x−h)_{2}+k$  $a,$ $h,$ and $k$ are real numbers, $a =0,$ and $(h,k)$ is the vertex of the parabola. 
Intercept Form (also called Factored Form) 
$y=a(x−x_{1})(x−x_{2})$  $a,$ $x_{1},$ and $x_{2}$ are real numbers, $a =0,$ and $x_{1}$ and $x_{2}$ are the $x$intercepts of the parabola. 
The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
A parabola either opens upward or downward. This is called its direction. If the leading coefficient $a$ of the corresponding equation is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.
If a function's axis of symmetry is the $y$axis, it is said that it is an even function.
A parabola can intersect the $x$axis at zero, one, or two points. Since the function's value at an $x$intercept is always $0,$ these points are called zeros, or sometimes roots.
Because all graphs of quadratic functions extend infinitely to the left and right, they each have a $y$intercept somewhere along the $y$axis.
Consider the given parabola. Identify its zeros, line of symmetry, or $y$intercept.
A quadratic function is said to be written in vertex form if it follows a specific format.
$y=a(x−h)_{2}+k$
Here, $a,$ $h,$ and $k$ are real numbers with $a =0.$ The value of $a$ gives the direction of the parabola. When $a>0,$ the parabola faces upward, and when $a<0,$ it faces downward. The vertex of the parabola lies at $(h,k),$ and the axis of symmetry is the vertical line with equation $x=h.$
Consider the graph of $y=41 (x−4)_{2}+8.$
Comparing the generic vertex form with the example function, the values of $a,$ $h,$ and $k$ can be identified.Direction  Vertex  Axis of Symmetry 

$a=41 $  $h=4$ and $k=8$  $h=4$ 
Since $41 $ is less than $0,$ the parabola opens downward.  The vertex is located at $(4,8).$  The axis of symmetry is the vertical line $x=4.$ 
Function $1$  Function $2$  Function $3$ 

$y=2(x+1)_{2}+7$ $⇕$ $y=2(x−(1))_{2}+7$ 
$y=(x−3)_{2}+1$ $⇕$ $y=1(x−3)_{2}+1$ 
$y=5(x−2)_{2}−3$ $⇕$ $y=5(x−2)_{2}+(3)$ 
When a quadratic function is written in vertex form, some characteristics of its graph can be identified.
$y=a(x−h)_{2}+k$  

Direction  Vertex  Axis of Symmetry 
$a>0a<0 ⇒upward⇒downward $

$(h,k)$  $x=h$ 
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is $x=h,$ here it is $x=2.$
LaShay loves playing golf.
She is trying to improve her swing by drawing the parabola that the ball will make. Using her math knowledge, she has calculated the quadratic function that corresponds to this parabola.The axis of symmetry of the parabola is the vertical line through the vertex. Therefore, in this case, the equation of the axis of symmetry is $x=23 .$
The $y$intercept will now be determined and plotted. To do so, $x=0$ will be substituted in the given equation.$x=0$
Subtract term
$(a)_{2}=a_{2}$
$(ba )_{m}=b_{m}a_{m} $
$a⋅cb =ca⋅b $
$ba =b/2a/2 $
Add fractions
Mark is studying parabolas so that he can help LaShay with her golf swing. He wants to write the vertex form of the quadratic function that corresponds to the given graph.
Help Mark find the desired equation!Start by identifying the vertex of the parabola.
$x=0$, $y=1$
Subtract term
$(a)_{2}=a_{2}$
$LHS+2=RHS+2$
$LHS/9=RHS/9$
$ba =b/3a/3 $
Rearrange equation
A quadratic function is said to be written in factored form or intercept form if it follows a specific format.
$y=a(x−p)(x−q)$
Here, $a,$ $p,$ and $q$ are real numbers with $a =0.$ The value of $a$ gives the direction of the parabola. When $a>0,$ the parabola faces upward, and when $a<0,$ it faces downward. The zeros of the parabola are $p$ and $q,$ and the axis of symmetry is the vertical line with equation $x=2p+q .$
Consider the graph of $y=21 (x−7)(x−13).$
Comparing the generic factored form with the example function, the values of $a,$ $p,$ and $q$ can be identified.Direction  Zeros  Axis of Symmetry 

$a=21 $  $p=7$ and $q=13$  $2p+q $ $⇓$ $27+13 =10$ 
Since $21 $ is greater than $0,$ the parabola opens upward.  The zeros are $7$ and $13.$ Therefore, the parabola intersects the $x$axis at $(7,0)$ and $(13,0).$  The axis of symmetry is the vertical line $x=10.$ 
Function $1$  Function $2$  Function $3$ 

$y=2(x+1)(x−3)$ $⇕$ $y=2(x−(1))(x−3)$ 
$y=(x−5)(x−9)$ $⇕$ $y=1(x−5)(x−9)$ 
$y=5x(x−2)$ $⇕$ $y=5(x−0)(x−2)$ 
In the following applet, several quadratic functions are expressed in different forms. Are the quadratic functions written in vertex form, factored form, or neither?
When a quadratic function is written in factored form, some characteristics of its graph can be identified.
$y=a(x−p)(x−q)$  

Direction  Zeros  Axis of Symmetry 
$a>0a<0 ⇒upward⇒downward $

$p$ and $q$  $x=2p+q $ 
It is possible to graph a quadratic function using these characteristics. Consider the function $y=(x+1)(x−5).$
$x=2$
Add and subtract terms
Remove parentheses
$a(b)=a⋅b$
By now, it is not a secret that LaShay loves playing golf.
She is once again trying to improve her swing by drawing the parabola that the ball will make. One more time, she uses her math knowledge to calculate the quadratic function that corresponds to this parabola.Note that, for the given function, the value of $a$ is $2.$ Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.
Continuing his studies, Mark wants to write the factored form of the quadratic function that corresponds to the given parabola.
Help Mark find the desired equation!Start by identifying the $x$intercepts of the parabola.
$x=2$, $y=3$
Add and subtract terms
$a(b)=a⋅b$
$LHS/(9)=RHS/(9)$
$ba =ba $
$ba =b/3a/3 $
Rearrange equation
For the following quadratic functions, identify the vertex or the zeros.
$(a−b)_{2}=a_{2}−2ab+b_{2}$
Commutative Property of Multiplication
Multiply
Calculate power
Distribute $2$
Distribute $(x−3)$
Distribute $2x&2$
Add terms