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Unit 3: Quadratic Functions
Definitions
Linear relation
: A relation with a constant rate of change, or with a constant first differenceQuadratic relation
: A relation where the 2nd differences are constantNon-linear relation
: A relation with a variable rate of changeDegree
of a polynomial: Highest exponent that appears in any term of a polynomiala-value
: Step propertyAxis of symmetry
: Vertical symmetrical line through a parabola, also x-value for vertexZeroes
: x-intercepts of parabolaVertex
: Maximum or minimum value on a parabolaOptimal value
: y-value for vertexDomain
: List of all valid x-values for relation, expressed as or as a relation such asRange
: List of all valid y-values for relation, expressed as or as a relation such asRelation
: Set of ordered pairs of numbersFunction
: A relation in which no y-coordinates share an x-coordinate (e.g., circles are not functions)
Parabolas
A standard graph of a parabola 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 where .
- By using factored form, we can easily see that an equation with factored form of has its x-intercepts at .
Y-intercepts
- These are the values of where .
Vertex
- This highest/lowest value of that the parabola takes.
- This point tells us alot of things, including the
axis of symmetry
and themaximum/minimum/optimal
value. - With vertex form in , we know that the vertex is at .
Optimal value
- This is basically the 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 , the axis of symmetry is at
- Using standard form , the axis of symmetry is at
- Using vertex form, , the axis of symmetry is simply the coordinate of the vertex, so .
Direction Of Opening
- To put it bluntly, the parabola opens upward if the value is positive, and downwards if the value is negative.
- Think of a postive value as a happy face, and a negative 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 value of is for any . The step property for any other a value is for any .
- The step property tells us the difference of values between each
point starting from the vertex, meaing if the vertex is at ,
the next two points would be , where is the 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 for any has more than point on the function, then that relation is not a function. Bluntly, a function cannot have a vertical straightline touching any of its points.
Ways of Representing Functions
1. Table Of Values
2. Coordinates in a Set
3. Graph
4. Mapping (Bubble Diagrams)
5. Equation
6. Function Notation
Domain And Range
- A
domain
is the set of -values, and therange
is the set of -values. - To represent domain and range, we use set notation to represnt it or simply by listing the and values.
- We use listing method for representing points, such as:
- Make sure the values or sorted in order.
- We use set notation or interval
notation to represent a continous graph, such as:
Transformations
There are 3 main types of transformations for a quadratic function.
Vertical Translation
- When we graph the quadratic relation , the vertex of the parabola has coordinates
- When , the graph of is vertically translated up units.
- When , the graph of is vertically translated down units.
Horizontal Translation
- When we graph the quadratic relation, the vertex of the parabola has coordinates
- When , the graph of is horizontally translated left units.
- When , the graph of is horizontally translated right units.
Vertical Stretch/Compression
- If , the graph is reflected over the x-axis.
- If vertical stretch.expansion by a factor of .
- If vertical compression by a factor of
- The step property also gets affected.
Forms of Quadratic Functions
- A quadratic relation in the form is said to be in factored form. The zeroes are and . -The axis of symmetry can be determined by using the formula -The axis of symmetry is also x-coordinate of the vertex.
Quadratic Form | Function | Zeroes | Vertex | Axis of Symmetry |
---|---|---|---|---|
standard | Plug in Axis of symmetry and solve for | |||
vertex | Set , solve for | |||
factored | Plug in Axis of symmetry and solve for |
- Notice that the value stays the same in all the forms.
- By using completing the square, we can find that the vertex is at , where
Partial Factoring
- If cannot be factored, then we can use partial facotring to determine the vertex.
- We set , then we basically now stated that . Since there is no constant value, we can factor the equation to becoming , from where we can solve for the values of .
- In respect of the axis of symmetry, using the values we can find the axis of symmetry, by using the formula , 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 , then it would be denoting the values that are above the x-axis, and below the y-axis for the opposite.