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

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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,x2,x3`D = {x, x_2, x_3}` or as a relation such as D={xR,x=,x<50}`D = \{x \in \mathbb{R}, x =\not 0, x < 50\}`
  • Range: List of all valid y-values for relation, expressed as R=y,y2,y3`R = {y, y_2, y_3}` or as a relation such as R={yR,y=,y<50}`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=x2`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`x` where y=0`y=0`.
  • By using factored form, we can easily see that an equation with factored form of y=a(xr)(xs)`y=a(x-r)(x-s)` has its x-intercepts at (r,0),(s,0)`(r, 0), (s, 0)`.

Y-intercepts

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

Vertex

  • This highest/lowest value of y`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(xh)2+k`y=a(x-h)^2 + k`, we know that the vertex is at (h,k)`(h, k)`.

Optimal value

  • This is basically the y`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(xr)(xs)`y=a(x-r)(x-s)`, the axis of symmetry is at r+s2`\dfrac{r+s}{2}`
    • Using standard form y=ax2+bx+c`y=ax^2+bx+c`, the axis of symmetry is at b2a`\dfrac{-b}{2a}`
    • Using vertex form, y=a(xh)2+k`y=a(x-h)^2+k`, the axis of symmetry is simply the x`x` coordinate of the vertex, so h`h`.

Direction Of Opening

  • To put it bluntly, the parabola opens upward if the a`a` value is positive, and downwards if the a`a` value is negative.
  • Think of a postive a`a` value as a happy face, and a negative a`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`a` value of 1`1` is 1,3,5,7,2n+1`1, 3, 5, 7, \cdots 2n+1` for any n1`n \ge 1`. The step property for any other a value is 1a,3a,5a,7a,a(2n+1)`1a, 3a, 5a, 7a, \cdots a(2n+1)` for any n1`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)`(x, y)`,
    the next two points would be (x1,y+n),(x+1,y+n)`(x-1, y+n), (x+1, y+n)`, where n`n` is the nth`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`x=n` for any n`n` has more than 1`1` point on the function, then that relation is not a function. Bluntly, a function cannot have a vertical straightline touching any 2`2` of its points.

Ways of Representing Functions

1. Table Of Values

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

2. Coordinates in a Set

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

3. Graph

4. Mapping (Bubble Diagrams)

5. Equation

y=x25`y=x^2-5`

6. Function Notation

f(x)=x25`f(x) = x^2 - 5`

Domain And Range

  • A domain is the set of x`x`-values, and the range is the set of y`y`-values.
  • To represent domain and range, we use set notation to represnt it or simply by listing the x`x` and y`y` values.
  • We use listing method for representing points, such as:
    • D={3,1,0,1,2,2.5}`D = \{-3, -1, 0, 1, 2, 2.5 \}`
    • R={2,1,0.5,3,3,5}`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={xxR}`D = \{x \mid x \in \mathbb{R} \}`
    • R={yy2,yR}`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=x2+k`y=x^2+k`, the vertex of the parabola has coordinates (0,k)`(0, k)`
  • When k>0`k \gt 0`, the graph of y=x2`y=x^2` is vertically translated up k`\mid k \mid` units.
  • When k0`k \le 0`, the graph of y=x2`y=x^2` is vertically translated down k`\mid k \mid` units.

Horizontal Translation

  • When we graph the quadratic relation, y=(xh)`y=(x-h)` the vertex of the parabola has coordinates (h,0)`(h, 0)`
  • When h>0`h \gt 0`, the graph of y=x2`y=x^2` is horizontally translated left k`\mid k \mid` units.
  • When h0`h \le 0`, the graph of y=x2`y=x^2` is horizontally translated right k`\mid k \mid` units.

Vertical Stretch/Compression

  • If a<0`a \lt 0 \rightarrow`, the graph is reflected over the x-axis.
  • If agt1`\mid a \mid gt 1 \rightarrow` vertical stretch.expansion by a factor of a`\mid a \mid`.
  • If 0<a<1`0 \lt \mid a \mid \lt 1 \rightarrow` vertical compression by a factor of a`\mid a \mid`
  • The step property also gets affected.

Forms of Quadratic Functions

  • A quadratic relation in the form y=a(xr)(xs)`y=a(x-r)(x-s)` is said to be in factored form. The zeroes are x=r`x=r` and x=s`x=s`. -The axis of symmetry can be determined by using the formula r+s2`\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=ax2+bx+c`y=ax^2+bx+c` x=b±b24ac2a`x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}` Plug in Axis of symmetry and solve for y`y` x=b2a`x = \dfrac{-b}{2a}`
vertex y=a(xh)2+k`y=a(x-h)^2+k` Set y=0`y=0`, solve for x`x` (h,k)`(h, k)` x=h`x = h`
factored y=a(xr)(xs)`y=a(x-r)(x-s)` (r,0),(s,0)`(r, 0), (s, 0)` Plug in Axis of symmetry and solve for y`y` x=r+s2`x = \dfrac{r+s}{2}`
  • Notice that the a`a` value stays the same in all the forms.
  • By using completing the square, we can find that the vertex is at (b2a,cb24a)`(\dfrac{-b}{2a}, c - \dfrac{b^2}{4a})`, where a=`a =\not 0`

Partial Factoring

  • If y=ax2+bx+c`y=ax^2+bx+c` cannot be factored, then we can use partial facotring to determine the vertex.
  • We set y=c`y =c`, then we basically now stated that 0=ax2+bx`0 = ax^2 + bx`. Since there is no constant value, we can factor the equation to becoming x(ax+b)`x(ax + b)`, from where we can solve for the values of x`x`.
  • In respect of the axis of symmetry, using the 2 x`2 \space x` values we can find the axis of symmetry, by using the formula r+s2`\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`0`, then it would be denoting the values that are above the x-axis, and below the y-axis for the opposite.