Quadratic Functions
How does the a, h, and k transform a parabola in vertex form?
(h, k) is the vertex of the parabola, and x = h is the axis of symmetry. • the h represents a horizontal shift (how far left, or right, the graph has shifted from x = 0). • the k represents a vertical shift (how far up, or down, the graph has shifted from y = 0).
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.
How do you know a function is 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.
Monomial
Algebraic expression consisting of one term
Binomial
Algebraic expression of the sum or difference of two terms
Trinomial
Algebraic expression of three terms
How do you graph the inverse of a quadratic function?
Determine which form of quadratic equation you have, Define your variable, Calculate h, Calculate k, Plot your vertex, Draw the parabola's axis, Find the direction of opening, if necessary, find and plot x intercepts, If necessary, find and plot the y intercept, If necessary, plot additional points, then graph
Why do we factor a trinomial?
Factoring is used to solve different kinds of problems.
How do you graph a quadratic f(x)?
To find the x-value of the vertex x = -b/2a , we would substitute -12 in for "b" and 2 in for "a", to get x = - (-12)/(2(2)) = 3. So our x-value of the vertex is x = 3
How do you find the y-intercept of a quadratic function?
To find the y-intercept, substitute 0 in for x and solve for f(x): f(0) = (0)2 - 3(0) + 5.
Give three ways to find the x-intercepts of a quadratic function.
Using a Graph of a Line Using the Quadratic Formula & Using the Equation of the Line
Steps to Factoring a Trinomial
identify a,b, and c in the trinomial ax2 + bx+c. write down all factor pairs of c. identify which factor pair from the previous step sums up to b. Substitute factor pairs into two binomials.
How do you find the domain of a parabola?
look for the left-most and right-most x-values on the graph
How do you find the range of a parabola?
look for the lower-most and upper-most y-values on the graph
Does a quadratic function have any holes?
no
X-Intercepts of a Quadratic
substitute 0 in for y and solve for x.
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.
Quadratic Formula
the solution to the quadratic equation
How do you find the intervals of a parabola?
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
How do you find the vertex of a quadratic function that is in vertex form?
If 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.
Does a quadratic have horizontal or vertical asymptotes?
None
What is the parent function of a quadratic?
The graph of any quadratic function is referred to as a parabola. shall be called the "parent" graph for all quadratic functions
What is the purpose of the quadratic formula?
The quadratic formula is a tool. A quadratic equation is usually given to you specifically for the purpose of solving it. When solving a quadratic equation, you are finding the roots of that equation. The roots of the equation will tell you where the graph touches or crosses the x - axis.
How do you graph the parent function of a quadratic?
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.
Steps for the Quadratic Formula
The 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.
How do you convert a quadratic function in standard form to vertex 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.
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.
Describe what the maximum of a parabola is.
When the parabola opens down, the vertex is the highest point on the graph
Describe what the minimum of a parabola is.
When the parabola opens up, the vertex is the lowest point on the graph
Standard Form of a Quadratic Function
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.
What are asymptotes?
a line that continually approaches a given curve but does not meet it at any finite distance.
What are holes in a function?
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.
Polynomial
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s)
Vertex Form of a Quadratic Function
f (x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola
How do you find the inverse equation of a quadratic function?
f 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)