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.