Quadratic Functions

How do you graph the inverse of a quadratic function?

) Replace f(x) by y 2) Switch the roles of "x" and "y" , in other words, interchange x and y in the equation. 3) Solve for y in terms of x 4) Replace y by f −1(x) to get the inverse function 5) *Sometimes, it is helpful to use the domain and range of the original function to identify the correct inverse function out of two possibilities. This happens when you get a "plus or minus" case in the end.

Factors of a Quadratic

A "quadratic" is a polynomial that looks like "ax2 + bx + c", where "a", "b", and "c" are just numbers. For the easy case of factoring, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance:

What is the parent function of a quadratic

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape.

Steps to Factoring a Trinomial

Define the model: Write how your system behaves (x^2 + x) Define the desired state: What should it equal? (6) Define the error: The error is its own system: Error = actual - desired (i.e., x^2 + x - 6) Factor the error: Rewrite the error as interlocking components: (x + 3)(x - 2) Reduce the error to zero: Zero out one component or the other (x = -3, or x = 2).

How do you find the y-intercept of a quadratic function?

Find the x-intercept(s). To find the x-intercept let y = 0 and solve for x. You can solve for x by using the square root principle or the quadratic formula (if you simplify the problem into the correct form). Step 4: Graph the parabola using the points found in steps 1 - 3.

How do you find the inverse equation of a quadratic function?

If the domain lies to the right of the stationary point i.e. x > a certain value, use the + sign. Then, make x the subject of the formula. Replace y with x, and x with f-1(x), and congratulate yourself on having successfully found the inverse of a quadratic function.

Quadratic Formula

In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing.

What are holes in a function?

In general, a vertical asymptote occurs in a rational function at any value of x for which the denominator is equal to 0, but for which the numerator is not equal to 0.

How does the a, h, and k transform a parabola in vertex form?

a-if its going to be minimum or maximum h-if it goes left or right k-if it goes up or down

Monomial

an algebraic expression consisting of one term A monomial is an expression in algebra that contains one term, like 3xy. Monomials include: numbers, whole numbers and variables that are multiplied together, and variables that are multiplied together.

Binomial

an algebraic expression with two terms The only time you will get a binomial back as an answer is if both of your binomials share like terms like this: Our first binomial is 5x+3y, and our second binomial is 4x+7y. The first term of both binomials have the same variable to the same exponent, x.

Polynomial

an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable. 5x +1 Since all of the variables have integer exponents that are positive this is a polynomial. (x7 + 2x4 - 5) * 3x Since all of the variables have integer exponents that are positive this is a polynomial. 5x-2 +1 Not a polynomial because a term has a negative exponent 4 more rows

Trinomial

an expression with three terms, Factoring Trinomials. A trinomial is a 3 term polynomial. For example, 5x2 − 2x + 3 is a trinomial. In many applications in mathematics, we need to solve an equation involving a trinomial.

How do you find the vertex of a quadratic function that is in vertex form?

f the equation is y = 3(x + 4)2 - 6, the value of h is -4, and k is -6. To convert a quadratic from y = ax2 + bx + c form to vertex form, y = a(x - h)2+ k, you use the process of completing the square. Let's see an example. Convert y = 2x2 - 4x + 5 into vertex form, and state the vertex.

Steps for the Quadratic Formula

he solution of a quadratic equation is the value of x when you set the equation equal to zero. ... Given a quadratic equation: ax ² + bx + c. Quadratic Equation: y = x² + 2x + 1, a = 1, b = 2, c = 1. Below is a picture representing the graph of y = x² + 2x + 1 and its solution.

Give three ways to find the x-intercepts of a quadratic function.

identify the xaxis find the point where the line crosses the the x axis write the ordered pairs for the x intercept

Vertex Form of a Quadratic Function

minimum or maximum of an equation when it is graphed .

Where is the minimum or maximum of parabola located?

the minimum or maximum value of a parabola by identifying the y-coordinate of the vertex. Take a look at this graph. The vertex is located at the point (2.5, -.5), and the parabola opens up. That means that the parabola has a minimum value, which is y = 2.5.

How do you convert a quadratic function in standard form to vertex form?

to convert a quadratic form y=ax2+bx+c form to vertex from , y=a(x-h)2+k , you use the process of completing the square . lets see an example. convert y=2x2-4x+5 into vertex form .

Does a quadratic have horizontal or vertical asymptotes?

vertical

X-Intercepts of a Quadratic

x-intercepts of any equation, substitute 0 in for y and solve for x. So, we have 0 = 3x2+ x + 1. Now, use the quadratic equation to solve for x, wich a = 3, b = 1, and c = 1: So, now we can find the value of the x-intercepts

How do you find the intervals of a parabola?

When x<c, the value of y will increase (as x increases). If your parabola's vertex is at the point ( a , b ) , then it can be put into the form y = ( x − a ) 2 + b or y = − ( x − a ) 2 + b . In the former case, the parabola is decreasing on ( − ∞ , a ) and increasing on the interval ( a , ∞ )

Describe what the maximum of a parabola is.

^^^^^^^^

What are asymptotes?

a line that continually approaches a given curve but does not meet it at any finite distance.

Describe what the minimum of a parabola is.

In the line of symmetry discussion, we dealt with the x-coordinate of the vertex; and just like clockwork, we need to now examine the y-coordinate. The y-coordinate of the vertex tells us how high or how low the parabola sits. Once again with our trusty example, y = (x-3)2 + 4, we see that the y-coordinate of the vertex (as derived from the number on the far right of the equation) dictates how high or low on the coordinate plane that the parabola sits. This parabola is resting on the line y = 4 (see line of symmetry for why the equation is y = __, instead of x = __ ). Once we have identified what the y-coordinate is, the last question we have is whether this number represents a maximum or minimum. We call this number a maximum if the parabola is facing downward (the vertex represents the highest point on the parabola), and we can call it a minimum if the parabola is facing upward (the vertex represents the lowest point on the parabola).

Why do we factor a trinomial?

Learning to "factor an equation" is the process of arranging your teepee. In this case: x^2 + x - 6 &= (x + 3)(x -2) If x = -3 then Component A falls down. If x = 2, Component B falls down. Either value causes the error to collapse, which means our original system (x^2 + x, the one we almost forgot about!) meets our requirements: When x = -3, the error collapses, and we get (-3)2 + -3 = 6 When x = 2, the error collapses, and we get 22 + 2 = 6

Does a quadratic function have any holes?

No , it does not

How do you graph a quadratic f(x)?

Practice graphing the equation by plotting the vertex and the y-intercept as shown below. You may want to plot other points, also. Remember, you can pick any number to substitute in the equation for x and solve for y, and the corresponding point will be on the graph. So, plot the vertex, the y-intercept (0, 2).

How do you know a function is a quadratic?

Quadratic Equation in Standard Form: ax2 + bx + c = 0. Quadratic Equations can be factored. Quadratic Formula: x = −b ± √(b2 − 4ac) 2a. When the Discriminant (b2−4ac) is: positive, there are 2 real solutions. zero, there is one real solution. negative, there are 2 complex solutions.

Standard Form of a Quadratic Function

Quadratic function: is a function that can be written in the form f(x) = ax2 + bx + c where a, b, and c are real numbers and a = 0. Parabola: The graph of a squaring function is called a parabola. It is a U-shaped graph. Vertex of a parabola: The point on the parabola where the graph changes direction.

What is the purpose of the quadratic formula?

The Quadratic Formula uses the "a", "b", and "c" from "ax2 + bx + c", where "a", "b", and "c" are just numbers; they are the "numerical

How do you graph the parent function of a quadratic?

The function y=x2 or f(x) = x2 is a quadratic function, and is the parent graph for all other quadratic functions. The shortcut to graphing the function f(x) = x2 is to start at the point (0, 0) (the origin) and mark the point, called the vertex.

How do you find the domain of a parabola?

The values of a, b, and c determine the shape and position of the parabola. The domain of a function is the set of all real values of x that will give real values for y. The range of a function is the set of all real values of y that you can get by plugging real numbers into x. The quadratic parent function is y = x2.

How do you find the range of a parabola?

The values of a, b, and c determine the shape and position of the parabola. The domain of a function is the set of all real values of x that will give real values for y. The range of a function is the set of all real values of y that you can get by plugging real numbers into x. The quadratic parent function is y = x2. ( basically the same as domain)

How do you find the vertex of a quadratic function that is in standard form?

To convert a quadratic from y = ax2 + bx + c form to vertex form, y = a(x - h)2+ k, you use the process of completing the square. Let's see an example. Convert y = 2x2 - 4x + 5 into vertex form, and state the vertex.