Quadratic Functions

Factors of a Quadratic

"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: Factor x2 + 5x + 6.

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

How do you convert a quadratic function in standard form to 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

How do you find the intervals of a parabola?

If 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 , ∞ )

Describe what the maximum of a parabola is.

Maximum Value: The maximum value of a quadratic function f(x) = ax2 + bx + c where a < 0,="" is="" the="" y-="" coordinate="" of="" the="">

Describe what the minimum of a parabola is.

Minimum Value: The minimum value of a quadratic function f(x) = ax2 + bx + c where a > 0, is the y- coordinate of the vertex.

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.

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

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. Now, let's refer back to our original graph, y = x2, where "a" is 1.

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

Step 3: 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 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.

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. Hence, the vertex is given by (1,-3). We may also view the graph to verify our findings.

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. Below is a picture representing the graph of y = x² + 2x + 1 and its solution.<

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.

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

The xx-intercepts of the function f(x)=ax2+bx+c, a≠0f(x)=ax2+bx+c, a≠0 are the solutions of the quadratic equation ax2+bx+c=0ax2+bx+c=0. The solutions of a quadratic equation of the form ax2+bx+c=0ax2+bx+c=0 are given by the quadratic formula. x=−b±b2−4ac√2ax=−b±b2−4ac2a Here, the expression b2−4acb2−4ac is called the discriminant

X-Intercepts of a Quadratic

To find the 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 and not have to estimate!

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.

How do you know a function is a quadratic?

When written 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). • notice that the h value is subtracted in this form, and that the k value is added. If the equation is y = 2(x - 1)2 + 5, the value of h is 1, and k is 5. If the equation is y = 3(x + 4)2 - 6, the value of h is -4, and k is -6.

Vertex Form of a Quadratic Function

When written 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). • notice that the h value is subtracted in this form, and that the k value is added. If the equation is y = 2(x - 1)2 + 5, the value of h is 1, and k is 5. If the equation is y = 3(x + 4)2 - 6, the value of h is -4, and k is -6.

Binomial

an algebraic expression of the sum or the difference of two terms.

Trinomial

consisting of 3 terms

Monomial

consisting of one term

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.

Polynomial

more than 3 terms

Does a quadratic function have any holes?

no they do not

Quadratic Formula

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.

Does a quadratic have horizontal or vertical asymptotes?

vertical asymptotes