Quadratic Functions
Quadratic functions are functions that can be written in the standard form f(x)= ax^2 +bx +c, where "a" does not equal 0.
The simplest form of this, or the quadratic parent function is y= x^2
The domain of a quadratic function is all possible x-values of a graph, that when plugged into the function, make it true.
The range of a quadratic function is all possible y-values of a graph, that when plugged into a function, make it true.
If you graph the parent function, the domain will be all real numbers and the range will be anything more than or equal zero.
When the formula is graphed, it makes a parabola, or a horse-shoe shaped line.
The simplest form of this, or the quadratic parent function is y= x^2
The domain of a quadratic function is all possible x-values of a graph, that when plugged into the function, make it true.
The range of a quadratic function is all possible y-values of a graph, that when plugged into a function, make it true.
If you graph the parent function, the domain will be all real numbers and the range will be anything more than or equal zero.
When the formula is graphed, it makes a parabola, or a horse-shoe shaped line.
Standard Form
The standard form of a function is as stated above, f(x)= ax^2 +bx +c
The vertex of this formula can be found by plugging numbers into this: (-b/2a, f(-b/2a)). This way, the axis of symmetry is equal to -b/2a. When x is zero, the y axis will be the axis of symmetry.
When you are working with the quadratic formula, the simpler form of it is standard form. It is essential to make sure that a quadratic formula can be transformed into standard form to ensure you have the correct values for a, b, and c. If you don't do this, you may get the wrong a, b, and c values and come out with a wrong solution.
The vertex of this formula can be found by plugging numbers into this: (-b/2a, f(-b/2a)). This way, the axis of symmetry is equal to -b/2a. When x is zero, the y axis will be the axis of symmetry.
When you are working with the quadratic formula, the simpler form of it is standard form. It is essential to make sure that a quadratic formula can be transformed into standard form to ensure you have the correct values for a, b, and c. If you don't do this, you may get the wrong a, b, and c values and come out with a wrong solution.
Vertex Form
The vertex form of a quadratic function is f(x)= a(x-h)^2 + k
The vertex form of a quadratic function is convenient when you want to find the vertex of the function. The terms "h" and "k" in vertex form, make up the vertex, which is (h, k). The vertex of a quadratic function is the minimum or maximum point of a parabola.
The vertex form of a quadratic function is convenient when you want to find the vertex of the function. The terms "h" and "k" in vertex form, make up the vertex, which is (h, k). The vertex of a quadratic function is the minimum or maximum point of a parabola.
Types of Translations
A parabola's vertex is not always the origin. In some cases, the vertex is shifted, or translated. Translations that a parabola can go through include horizontal, vertical, dilations, and reflections.
horizontal translations: The "h" in vertex determines how far right or left the vertex moves. If the number is negative, you move it to the right however many units. If the number is positive, you move it to the left however many units it says.
For Example: the formula is, in vertex form, y= (x-1)^2 +2 As far as horizontal goes, the parabola will move to the right one.
Vertical translations: The "k" in vertex form determines how far up or down the vertex moves. If the number is negative, you move it down however many units. If the number is positive, you move it up however many units.
For Example: the formula is, in vertex form, y= (x-1)^2 +2 As far as vertical goes, the parabola will move up two units.
Dilations: The "a" value determines how wide or skinny a parabola is. If the a value is more than 1,then parabola will be skinnier than the graphed parent function. If the a value is more than zero but less than 1,then the parabola will be wider than the graphed parent function.
For Example:
horizontal translations: The "h" in vertex determines how far right or left the vertex moves. If the number is negative, you move it to the right however many units. If the number is positive, you move it to the left however many units it says.
For Example: the formula is, in vertex form, y= (x-1)^2 +2 As far as horizontal goes, the parabola will move to the right one.
Vertical translations: The "k" in vertex form determines how far up or down the vertex moves. If the number is negative, you move it down however many units. If the number is positive, you move it up however many units.
For Example: the formula is, in vertex form, y= (x-1)^2 +2 As far as vertical goes, the parabola will move up two units.
Dilations: The "a" value determines how wide or skinny a parabola is. If the a value is more than 1,then parabola will be skinnier than the graphed parent function. If the a value is more than zero but less than 1,then the parabola will be wider than the graphed parent function.
For Example:
Reflections: The "f(x)" in vertex form determines if a parabola is reflected over the x or y axis. If there is a negative in front of the f(x), like this, -f(x), then the parabola is reflected over the x-axis. If there is a negative in front of the x in f(x), like this, f(-x), then the parabola is reflected over the y-axis.
*This video will better explain all of the shifts. It refers to dilations as "stretch." It also refers to reflections as "flips."
*This video will better explain all of the shifts. It refers to dilations as "stretch." It also refers to reflections as "flips."
Notable Points
There is more to a parabola than the horse-shoe shaped line. There is the axis of symmetry, vertex, minimum and maximum values, y-intercept, and the roots.
Axis of Symmetry: This is the line that goes through the vertex and down the middle of the parabola. It acts as a mirror and reflects both sides of the parabola, so that they are the same. The value of x in the vertex determines where the axis of symmetry is, if x is 4, then the vertical line x=4 is the axis of symmetry.
Vertex: The vertex is the lowest or highest point in a parabola. It is also where the axis of symmetry goes through.
Minimum and Maximum values: The y value in the vertex represents the highest point, maximum, or lowest point, minimum, of a parabola.
Y-intercept: This is the where the parabola intersects the y-axis.
Roots: Roots are the points at which the parabola intersects the x-axis. If the vertex's x is zero there is only one root, but if not, then there will be two roots. If you solve the quadratic formula for x, you will get the roots of the parabola.
Axis of Symmetry: This is the line that goes through the vertex and down the middle of the parabola. It acts as a mirror and reflects both sides of the parabola, so that they are the same. The value of x in the vertex determines where the axis of symmetry is, if x is 4, then the vertical line x=4 is the axis of symmetry.
Vertex: The vertex is the lowest or highest point in a parabola. It is also where the axis of symmetry goes through.
Minimum and Maximum values: The y value in the vertex represents the highest point, maximum, or lowest point, minimum, of a parabola.
Y-intercept: This is the where the parabola intersects the y-axis.
Roots: Roots are the points at which the parabola intersects the x-axis. If the vertex's x is zero there is only one root, but if not, then there will be two roots. If you solve the quadratic formula for x, you will get the roots of the parabola.
Quadratic Formula and Uses for it
Unlike other solution methods, this formula can be used to solve any quadratic equation. This would be why it is used so often. You can you the quadratic formula to:
- calculate areas
- figure out a profit
- solve equations found in athletics
- find a speed
- solve equations in your math class ;)