Quadratic Functions
How do you find the vertex of a quadratic function that is in standard form?
If we can determine the values of the x intercepts, we can use symmetry to determine the x coordinate of the vertex. The x coordinate of the vertex will be the center of the two intercepts
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 do you graph the inverse of a quadratic function?
Key steps 1) 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.
Does a quadratic have horizontal or vertical asymptotes?
Quadratic functions do not have asymptotic behavior. Also, Quadratic functions are not Rational functions.
How do you find the inverse equation of a quadratic function?
Replace y with x, and x with f-1(x), and congratulate yourself on having successfully found the inverse of a quadratic function.
Steps for the Quadratic Formula
STEP 1: Divide equation by whatever is multiplied on the squared term. STEP 2: Keep all terms containing x on one side. Move the constant to the right. STEP 3: Take half of the x-term coefficient and square it. Add this value to both sides. STEP 4: Simplify right side STEP 5: Write the perfect square on the left. STEP 6: Take the square root of both sides. STEP 7: Solve for x
How do you graph a quadratic f(x)?
So, given a quadratic function, y = ax2 + bx + c, when "a" is positive, the parabola opens upward and the vertex is the minimum value. On the other hand, if "a" is negative, the graph opens downward and the vertex is the maximum value
X-Intercepts of a Quadratic
So, if we can't solve for x, that means there are no x-intercepts. Let's graph the parabola using the y-intercept (0, 5) and the vertex (3/2, 11/4). Remember, the parabola should not cross the x-axis anywhere. Also, remember since "a" is positive, the graph should open upward.
How do you find the domain of a 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 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.
Describe what the minimum of a parabola is.
The minimum value of a quadratic function f(x) = ax2 + bx + c where a > 0, is the y- coordinate of the vertex.
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 find the range of a parabola?
The range of a function is the spread of possible y-values (minimum y-value to maximum y-value) Substitute different x-values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?) Make sure you look for minimum and maximum values of y. Draw a sketch! In math, it's very true that a picture is worth a thousand words.
Give three ways to find the x-intercepts of a quadratic function.
The x intercepts of the graph of a quadratic function f given by f(x) = a x 2 + b x + c are the real solutions, if they exist, of the quadratic equation a x 2 + b x + c = 0 The above equation has two real solutions and therefore the graph has x intercepts when the discriminant D = b^2 - 4 a c is positive. It has one repeated solution when D is equal to zero. The solutions are given by the quadratic formulas
How does the a, h, and k transform a parabola in vertex form?
This form of a quadratic is useful when graphing because the vertex location is given directly by the values of h and k. In the graph above, click 'zero' under h and k, and note how the vertex is now at 0,0. The value of k is the vertical (y) location of the vertex and h the horizontal (x-axis) value. Move the sliders for h and k noting how they determine the location of the curve but not its shape. The value of a is the same as with the standard form - it determines the 'steepness' of the parabola and the sign of a determines if the curve open upwards or downwards. Positive values open at the top. Adjust the slider for a and see this for yourself. Be sure to try both positive and negative values
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 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. Let's see an example. Convert y = 2x2 - 4x + 5 into vertex form, and state the vertex.
How do you find the y-intercept of a quadratic function?
To find the y-intercept let x = 0 and solve for y.
Where is the minimum or maximum of parabola located?
We can identify 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.
Does a quadratic function have any holes?
When a value of x sets both the denominator and the numerator of a rational function equal to 0, there is a hole in the graph; that is, a single point at which the function has no value. f (x) = has a hole at x = 2 :
What are asymptotes?
a line that continually approaches a given curve but does not meet it at any finite distance.
Monomial
an algebraic expression consisting of one term
Binomial
an algebraic expression of the sum or the difference of two terms.
Trinomial
an algebraic expression of three terms.
Polynomial
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
What is the parent function of a quadratic?
ax2+bx+c
Steps to Factoring a Trinomial
ex. 6x^2 +19x -20 1) multiply the first coefficient by the constant (6)(-20)= -120 2) rewrite it to x^2 +19x -120 (notice that 6 is no longer there) 3) factor the new trinomial (changes to x^2 +bx +c) (x-5)(x+24) 4) add the first coefficient back to the equation (in front of x) (6x-5)(6x+24) 5) find the common factor of the numbers in each bracket ex. for the first bracket (6 and 5) there is no common factor so it's left as is for the second bracket (6 and 24) the common factor is 6 then divide each bracket by the common factor 6) result is (6x-5)(x+4)
How do you find the intervals of a parabola?
f your parabola's vertex is at the point (a,b) ( a , b ) , then it can be put into the form y=(x−a)2+b y = ( x − a ) 2 + b or y=−(x−a)2+b y = − ( x − a ) 2 + b . In the former case, the parabola is decreasing on (−∞,a) ( − ∞ , a ) and increasing on the interval (a,∞) ( a , ∞ ) .
How do you know a function is a quadratic?
f(x) = ax2 + bx + c
Standard Form of a Quadratic Function
f(x) = ax2 + bx + c
Factors of a Quadratic
he Quadratic Formula Explained. Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring.
Describe what the maximum of a parabola is.
the highest point of an upside down U shaped funtion
Why do we factor a Trinomial?
to find x-intercept
Quadratic Formula
x equals negative b plus or minus the square root of b squared minus four (ac) then divide by 2a
Vertex Form of a Quadratic Function
y = a(x - h)2+ k