highschool/Grade 10/Math/MPM2DZ/Unit 3: Quadratic Functions.md

Unit 3: Quadratic Functions

Definitions

• Linear relation: A relation with a constant rate of change, or with a constant first difference
• Quadratic relation: A relation where the 2nd differences are constant
• Non-linear relation: A relation with a variable rate of change
• Degree of a polynomial: Highest exponent that appears in any term of a polynomial
• a-value: Step property
• Axis of symmetry: Vertical symmetrical line through a parabola, also x-value for vertex
• Zeroes: x-intercepts of parabola
• Vertex: Maximum or minimum value on a parabola
• Optimal value: y-value for vertex
• Domain: List of all valid x-values for relation, expressed as $`D = {x, x_2, x_3}`$ or as a relation such as $`D = \{x \in \mathbb{R}, x =\not 0, x < 50\}`$
• Range: List of all valid y-values for relation, expressed as $`R = {y, y_2, y_3}`$ or as a relation such as $`R = \{y \in \mathbb{R}, y =\not 0, y < 50\}`$
• Relation: Set of ordered pairs of numbers
• Function: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)

Parabolas

A standard graph of a parabola $`y=x^2`$ would look something like this:

There are several things you should know about a parabola: - X-intercepts - Y-intercepts - Vertex - Optimal value - Axis of Symmetry - Direction Of Opening - Step Property

X-intercepts

• These are the zeroes of the quadratic function, or the solutions you found when solving for a quadratic function in factored form.
• These are the values of $`x`$ where $`y=0`$.
• By using factored form, we can easily see that an equation with factored form of $`y=a(x-r)(x-s)`$ has its x-intercepts at $`(r, 0), (s, 0)`$.

Y-intercepts

• These are the values of $`y`$ where $`x=0`$.

Vertex

• This highest/lowest value of $`y`$ that the parabola takes.
• This point tells us alot of things, including the axis of symmetry and the maximum/minimum/optimal value.
• With vertex form in $`y=a(x-h)^2 + k`$, we know that the vertex is at $`(h, k)`$.

Optimal value

• This is basically the $`y`$ value of the vertex, and is useful for maximum/minimum word problems
• If the parabola is opening upwards, this is the minimum value. If the parabola is opening downards, this is the maximum value

Axis of Symmetry

• A vertical line of symmetry for the parabola.
• This can be determined in many ways:
  • Using factored form $`y=a(x-r)(x-s)`$, the axis of symmetry is at $`\dfrac{r+s}{2}`$
  • Using standard form $`y=ax^2+bx+c`$, the axis of symmetry is at $`\dfrac{-b}{2a}`$
  • Using vertex form, $`y=a(x-h)^2+k`$, the axis of symmetry is simply the $`x`$ coordinate of the vertex, so $`h`$.

Direction Of Opening

• To put it bluntly, the parabola opens upward if the $`a`$ value is positive, and downwards if the $`a`$ value is negative.
• Think of a postive $`a`$ value as a happy face, and a negative $`a`$ value as a sad face.

Step Property

• This is a property that can be used to quickly graph a quadratic function.
• The step property for an $`a`$ value of $`1`$ is $`1, 3, 5, 7, \cdots 2n+1`$ for any $`n \ge 1`$. The step property for any other a value is $`1a, 3a, 5a, 7a, \cdots a(2n+1)`$ for any $`n \ge 1`$.
• The step property tells us the difference of values between each point starting from the vertex, meaing if the vertex is at $`(x, y)`$,
  the next two points would be $`(x-1, y+n), (x+1, y+n)`$, where $`n`$ is the $`n^{th}`$ step property number.

Functions

Function: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)

Vertical Line Test (VLT)

We can easily tell if a relation is a function by using the vertical line test. If a single straight line of $`x=n`$ for any $`n`$ has more than $`1`$ point on the function, then that relation is not a function. Bluntly, a function cannot have a vertical straightline touching any $`2`$ of its points.

Ways of Representing Functions

1. Table Of Values

$`x`$	$`y`$
$`2`$	$`5`$
$`5`$	$`7`$
$`6`$	$`5`$

2. Coordinates in a Set

$`f = \{(-2, 1), (0, 1), (3, 1), (4, 1), (7, 1)\}`$

3. Graph

4. Mapping (Bubble Diagrams)

5. Equation

$`y=x^2-5`$

6. Function Notation

$`f(x) = x^2 - 5`$

Domain And Range

• A domain is the set of $`x`$-values, and the range is the set of $`y`$-values.
• To represent domain and range, we use set notation to represnt it or simply by listing the $`x`$ and $`y`$ values.
• We use listing method for representing points, such as:
  • 
  • $`D = \{-3, -1, 0, 1, 2, 2.5 \}`$
  • $`R = \{-2, -1, 0.5, 3, 3, 5 \}`$
  • Make sure the values or sorted in order.
• We use set notation or interval notation to represent a continous graph, such as:
  • 
  • $`D = \{x \mid x \in \mathbb{R} \}`$
  • $`R = \{y \mid y \ge 2, y \in \mathbb{R} \}`$

Transformations

There are 3 main types of transformations for a quadratic function.

Vertical Translation

• When we graph the quadratic relation $`y=x^2+k`$, the vertex of the parabola has coordinates $`(0, k)`$
• When $`k \gt 0`$, the graph of $`y=x^2`$ is vertically translated up $`\mid k \mid`$ units.
• When $`k \le 0`$, the graph of $`y=x^2`$ is vertically translated down $`\mid k \mid`$ units.

Horizontal Translation

• When we graph the quadratic relation, $`y=(x-h)`$ the vertex of the parabola has coordinates $`(h, 0)`$
• When $`h \gt 0`$, the graph of $`y=x^2`$ is horizontally translated left $`\mid k \mid`$ units.
• When $`h \le 0`$, the graph of $`y=x^2`$ is horizontally translated right $`\mid k \mid`$ units.

Vertical Stretch/Compression

• If $`a \lt 0 \rightarrow`$, the graph is reflected over the x-axis.
• If $`\mid a \mid gt 1 \rightarrow`$ vertical stretch.expansion by a factor of $`\mid a \mid`$.
• If $`0 \lt \mid a \mid \lt 1 \rightarrow`$ vertical compression by a factor of $`\mid a \mid`$
• The step property also gets affected.

Forms of Quadratic Functions

• A quadratic relation in the form $`y=a(x-r)(x-s)`$ is said to be in factored form. The zeroes are $`x=r`$ and $`x=s`$. -The axis of symmetry can be determined by using the formula $`\dfrac{r + s}{2}`$ -The axis of symmetry is also x-coordinate of the vertex.

Quadratic Form	Function	Zeroes	Vertex	Axis of Symmetry
standard	$`y=ax^2+bx+c`$	$`x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}`$	Plug in Axis of symmetry and solve for $`y`$	$`x = \dfrac{-b}{2a}`$
vertex	$`y=a(x-h)^2+k`$	Set $`y=0`$, solve for $`x`$	$`(h, k)`$	$`x = h`$
factored	$`y=a(x-r)(x-s)`$	$`(r, 0), (s, 0)`$	Plug in Axis of symmetry and solve for $`y`$	$`x = \dfrac{r+s}{2}`$

• Notice that the $`a`$ value stays the same in all the forms.
• By using completing the square, we can find that the vertex is at $`(\dfrac{-b}{2a}, c - \dfrac{b^2}{4a})`$, where $`a =\not 0`$

Partial Factoring

• If $`y=ax^2+bx+c`$ cannot be factored, then we can use partial facotring to determine the vertex.
• We set $`y =c`$, then we basically now stated that $`0 = ax^2 + bx`$. Since there is no constant value, we can factor the equation to becoming $`x(ax + b)`$, from where we can solve for the values of $`x`$.
• In respect of the axis of symmetry, using the $`2 \space x`$ values we can find the axis of symmetry, by using the formula $`\dfrac{r+s}{2}`$, since the axis of symmetry works for any 2 opposite points on the parabola.
• With this, we can easily find the vertex of a quadratic equation.

Quadratic Inequalities

• Quadratic inequalites can be solved graphically and algebraically. Since we know how to graph quadratic relations, we can solve quadtratic inequalites graphically
• An example of an algebraic solution of an quadratic inequality would be the number line method we learned in the previous unit.
• To solve it graphically, we will need to put the equation into factored form, then finding out the x-intercepts. With these, we can find out the side of the graph the solution is on.
• If the solution is greater than $`0`$, then it would be denoting the values that are above the x-axis, and below the y-axis for the opposite.