Quadratic equations

Discriminant equal to 0

1 real solution

Steps to graph quadratic function intercept form

1. Identify X intercept 2. Draw Axis of symmetry 3. Plot vertex (take X coordinate and step 2, plug into equation, solve for y) 4. Draw parabola-vertex and intercept points

Shorthand Rall to figure out the sign of the factors in the factored form of the quadratic equation

1. If variable c is negative the factors have different signs 2. If c is positive, look at variable b 3. If b is positive, signs are positive 4. If b is negative signs are negative

Name the 6 different forms of the quadratic equation

1. Parent form, y=x² 2. Standard form, y=ax²+bx+c 3. Intersect form, y = a(x-p)(x-q) 4. Factor form, y = a(x+p)(x+q) 4. y=ax²+c 5. y=ax²+bx 6. Vertex form, y = a(x-h)² + k

What are the 5 methods of solving a quadratic equation

1. Quadratic formula 2. Completing the square 3. Findings square roots 4. Graphing 5. Factoring

4 common forms of the equation of a line

1. Slope intercept form 2. Point slope form 3. Two-point equation form 4. Intercept form

Characteristics of the graph y = a(x-h)^2 + k

1. Vertex is the point (h,k) 2. Axis of symmetry is x = h 3. Graph opens up if a greater than 0, parabola opens down of a less than 0

Discriminant greater than 0

2 real solutions

2x^2 8x = 0 Solved by factoring

2x^2 8x = 0 2x (x + 4) = 0 2x = 0 or x = 4 = 0 Therefore x = 0 and x = -4

Solve using square roots 2x²= 8

2x²= 8 x²= 4 x = +2

How many different forms of the quadratic equation

6

An example using logarithms: e^xy = 7

An example using logarithms: e^xy = 7 ln(exy) = ln(7) xy = 1.9459

What happens to the signs of the variables p and q in the factored form of the quadratic equation : b is negative and c positive

Both signs are negative Both p and q are negative

What happens to the signs of the variables p and q in the factored form of the quadratic equation : b and c positive

Both signs are positive Both p and q are positive

y=ax²+bx Method to solve

Completing the square

y = a(x-p)(x-q) Method to solve

Factoring 0 products solution

Discriminant less than 0

No answer to the quad

What is the graph of the quadratic function look like

Parabola

How to use of for a polynomial equation in the factored form

Solve using the 0 product properly

The inverse of exponentiation

The inverse of exponentiation is called a root. The root operation yields a number which when multiplied by itself (n-1) times equals the original number. To be more clear, the inverse of the exponential operation y = x^n is the root operation x = nth root of y which can be expressed in words as "x is the nth root of y."

What happens to the graph of the parabola as the value of a decreases, a is less than 1 greater than 0 (a = 1/4)

The parabola gets wider

How do you calculate the variable c to complete the square?

add (b/2)^2 to both sides of the equation

Polynomials are a sum of terms based on power functions

f (x) = ax^4 + bx^3 + cx^2 + dx + e

Exponential functions take the form

f (x) = e^ax

Logarithmic functions are the inverse of exponential functions

f (x) = lnx = logex where e^lnx = x

parent quadratic function

the most basic quadratic function in the family of quadratic functions, y = x^2

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

y = a(x-h)^2 + k What is the Axis of symmetry

x = h

Complete the square of x²+5x

x² + 5x x² + 5x + (5/2)^2 x² + 5x + 25/4 = (x + 5/2)^2

Find vertex of y = -2x^2 + 12x -7

y = -2x^2 + 12x -7 x- coordinate of vertex x = -b/2a = 3 y-coordinate of vertex y = -2(3)^2 + 12(3) - 7 = 11 Vertex = (3,11)

Point slope form of line equation

y-y1=m(x-x1)

Characteristics of y = a(x-p)(x-q)

1. This formula represents the intercept form of the quadratic equation 2. The x intercepts are p and q 3. Axis of symmetry is halfway between p and q, Axis the symmetry is x = p + q/ 2 4. Parabola opens up if a is greater than 0 5. Parabola opens down if a is less than 0

An inequality can be expressed in 1 of 4 possible relationship

An inequality can express one of four possible relationships: greater than, less than, greater than or equal to, less than or equal to.

Graph of y = ax^2

As the value of a increases the parabola becomes more narrow

Graph of y = x^2 - c. What happens parabola as c increase

As the value of c decreases the graph of the parabola moves down on the Y axis

Graph of y = x^2 + c. What happens parabola as c increase

As the value of c increases the graph of the parabola moves up on the Y axis

Rules when manipulating an inequality

Cut to the Chase... Rules when simplifying inequalities. When manipulating an inequality by performing the same action to both sides of the inequality, you need to be careful of the effect on the sense or direction of the inequality. Follow these rules: 1. Add or subtract any value Same direction 2. Multiply or divide by a positive number Same direction 3. Multiply or divide by a negative number Reverse direction

Parabola opens in which direction if a is less than 0

Down

Inequality with one variable 7x − 2 ≤ 3x + 1

Example with one variable 7x − 2 ≤ 3x + 1 can be simplified 4x ≤ 3 x ≤ 3/4

Example with two inequalities 2 ≥ x ≥ − 1

Example with two inequalities 2 ≥ x ≥ − 1 In this example, x must satisfy both inequality relationships, so its valid range is from -1 to +2. Intersections

Example with two variables 7x + y ≤ 3x + 2y + 1

Example with two variables 7x + y ≤ 3x + 2y + 1 4x − y − 1 ≤ 0 y ≥ 4x − 1

y=ax²+c Method to solve

Findings Square roots

y=x² Method to solve

Findings Square roots

Solve for x, x^3/2 = 20

For example: x^3/2 = 20 (x^3/2)^2/3 = (20)^2/3 x = 7.368

y = a(x-h)² + k Method to solve

Graphing

0 product property

If the product of 2 equations equals 0. One or both of those equations must equals 0

Imaginary numbers

Imaginary numbers The concept of a square root of a negative number frustrated mathematicians for a long time. Did it really exist? Eventually, a solution was proposed -- just call these numbers imaginary! Just because they aren't real doesn't imply that they have no use. By defining the imaginary number i as i = −1, mathematicians were no longer limited to roots of positive numbers, and complex math was invented. It turns out that this can be very, very useful in real world problems. The function ejω, where j = −1 defines a sinusoidal AC signal in electrical engineering. Without imaginary numbers, your cell phone would never

Intersections

Intersections Similar to simultaneous equations, multiple inequalities that must be satisfied at the same time restrict the range of solutions to the intersection of the regions that satisfy each inequality individually. The region which satisfies the two inequalities y ≤ x + 2 2x ≤ − y + 4 is the green area in the graph.

Manipulating inequalities with a negative number

Manipulating or simplifying inequalities is similar to working with equations, with one very important difference. Multiplying or dividing both sides of an inequality by a negative number will reverse the sense or direction of the inequality. For example dividing both sides of this inequality by -4 −4x > 12 will change this from a greater than relationship to a less than relationship: x < − 3

Quadratic equation: Axis of symmetry of x

Quadratic equation y=ax²+bx+c Axis of symmetry x = -b/2a

y=ax²+bx+c Method to solve

Quadratic formula Factoring

Solve quadratic equation with no solution

Solution has no x-intercept

Solve (x - 4)(x + 2) = 0

Solve (x - 4)(x + 2) = 0 x-4 = 0 or x + 2 = 0 therefore x= 4 and x = -2

What technique can be used to add the constant c in a quadratic equation without a constant

Solve by completing the square

What happens to the signs of the variables p and q in the factored form of the quadratic equation : b and c negative

The signs are different when solving for the factor p and q have different signs

What happens to the signs of the variables p and q in the factored form of the quadratic equation : b is positive and c negative

The signs are different when solving for the factor p and q have different signs

Solve quadratic equation with 2 solution

The solution has 2 intercept

Solve quadratic equation with 1 solution

The solution has only one X intercept

Intercept form of line equation

This is used when the places where the line crosses the x and y axis are known. The y-intercept is where x is 0, and the x intercept is where y is 0. x/a + y/b = 1

Parabola opens in which direction if a is greater than 0

Up

How you solve quadratic equations in the following form y=ax²+c

Use square root ax²+c = 0

Quadratic equation: Vertex has an X coordinate

Vertex X x = -b/2a

y = a(x - h)^2 + k What are the points of the vertex

Vertex point is (h,k)

What is the square root of a negative number?

What happens when x is negative? Even integer exponents yield a positive result. ( − 2)^2 = 4 Odd integer exponents yield a negative result. ( − 2)^3 = − 8 Fractional exponents (roots) are imaginary. Yes, really. This is a mathematical concept where the square root of negative one is defined as the imaginary number 'i'. square root of −1 = i Real number, non-integer, exponents also yield imaginary results. ( − 2)^1.216

p + q in the factored form of the quadratic equation equal which variable

b

Discriminant of a quadratic equation

b^2-4ac

Solve using square roots b² + 12 = 5

b² + 12 = 5 b² = -7 NO SOLUTION Negative real numbers do not have a real square roots, therefore there is no solution

pq in the factored form of the quadratic equation equal which variable

c

Two-point equation form of line equation

two points on the line are known. y − y1 = y2 − y1/ x2 − x1 . (x − x1)

Complete the square of x² - 16x = -15

x² - 16x = -15 so, c = (b/2)² = -8² x² - 16x + (-8)² = -15 + (-8)² (x - 8)² = 49 x - 8 = +7 x = 8+7 x = 15, x = 1

Complete the square of x²+bx

x²+bx = x²+bx + (b/2)^2 = (x + b/2)^2

Quadratic function in vertex form

y = a(x-h)^2 + k

graphing of vertex for y = a(x-h)^2 + k

y = a(x-h)^2 + k graph of y = ax^2 The vertex of the parabola is moved horizontally h units The vertex of the parabola is moved vertically k unit

What are the x intercepts in the quadratic function, intercept form

y = a(x-p)(x-q) X intercepts are P and Q (p,0) and (q,0)

Quadratic function-intercept form

y = a(x-p)(x-q) a does not equal 0

Calculate Axis of symmetry using quadratic function and receptive form

y = a(x-p)(x-q) x = p + q/ 2

y-intercept in the Quadratic Equation

y = c, or (0, c) The point (0,c) is on the parabola

Slope intercept form of line equation

y = mx + b

How would you solve a quadratic equation in the following form y=ax²+bx

y=ax²+bx Add a constant c transformed into the standard form y=ax²+bx +c You can add the constant c by "completing the square"

quadratic function in standard form

y=ax²+bx+c A non-linear function that can be written as the above

y=ax²+bx+c, convert quadratic equation standard form into quadratic equation Factor form

y=ax²+bx+c standard quadratic form a(x + p)(x + q) Factored form p + q = b pq = c