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The vertex form of a quadratic function is given by
f ( x) = a( x - h) ² + k, where ( h, k) is the vertex of the parabola.

FYI: Different textbooks have different interpretations of the reference " standard form" of a quadratic function. Some say f ( x) = ax ² + bx + c is "standard form", while others say that f ( x) = a( x - h) ² + k is "standard form". To avoid confusion, this site will not refer to either as "standard form", but will reference f ( x) = a( x - h) ² + k as "vertex form" and will reference f(x) = ax ² + bx + c by its full statement.

To Convert from f ( x) = ax ² + bx + c Form to Vertex Form:
Method 1: Completing the Square
To convert a quadratic from y = ax ² + 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 = 2 x ² - 4 x + 5 into vertex form, and state the vertex.

Equation in y = ax ² + bx + c form.

Since we will be "completing the square" we will isolate the x ² and x terms ... so move the + 5 to the other side of the equal sign.

We need a leading coefficient of 1 for completing the square ... so factor out the current leading coefficient of 2.

Get ready to create a perfect square trinomial. BUT be careful!! In previous completing the square problems with a leading coefficient not 1, our equations were set equal to 0. Now, we have to deal with an additional variable, "y" ... so we cannot "get rid of " the factored 2. When we add a box to both sides, the box will be multiplied by 2 on both sides of the equal sign.

Find the perfect square trinomial. Take half of the coefficient of the x-term inside the parentheses, square it, and place it in the box.

Simplify and convert the right side to a squared expression.

Isolate the y-term ... so move the -3 to the other side of the equal sign.

In some cases, you may need to transform the equation into the "exact" vertex form of
y = a( x - h) ² + k, showing a "subtraction" sign in the parentheses before the h term, and the "addition" of the k term. (This was not needed in this problem.)

Vertex form of the equation.(The vertex of this graph will be moved one unit to the right and three units up from (0,0), the vertex of its parent y = 2( x - 1) ² + 3

Vertex = ( h, k) = (1, 3) y = x ².)

Here's a sneaky, quick tidbit:

When working with the vertex form of a quadratic function,

and

.
The " a" and " b" referenced here refer to f ( x) = a x ² + b x + c.

y = ax ² + bx + c form of the equation.

Find the vertex, ( h, k).

and

.
[ f ( h) means to plug your answer for h into the original equation for x.]

Write the vertex form.
y = a( x - h) ² + k

To Convert from Vertex Form to y = ax ² + bx + c Form:

Simply multiply out and combine like terms:

y = 2( x - 1) y = 2( y = 2 x ² - 4 x + 5 x ² - 2 x + 1) + 3
y = 2 x ² - 4 x + 2 + 3 ² + 3

Graphing a Quadratic Function in Vertex Form:

Start with the function in vertex form 1. :
y = a( x - h) ² + k

Pull out the values for If necessary, rewrite the function so you can clearly see the 2. h and k . h and k values.

(h, k ) is the vertex of the parabola.
Plot the vertex.

= -4 h = 2; k
Vertex: (2, -4)

. 3. The line x = h is the axis of symmetry
Draw the axis of symmetry.

x = 2 is the axis of symmetry

Find two or three points on one side of the axis of symmetry, by substituting your chosen 4. x-values into the equation.

For this problem, we chose (to the left of the axis of symmetry):

x = 1; y = 3( 1 - 2) ² - 4 = -1

x = 0; y = 3(0 - 2) ² - 4 = 8

Plot (1, -1) and (0,8)

5. Plot the mirror images of these points across the axis of symmetry, or plot new points on the right side. Remember, when drawing the parabola to avoid "connecting the dots" with straight line segments. A parabola is curved, not straight, as its slope is not constant.
Draw the parabola.

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Vertex Form of Quadratic Equation - MathBitsNotebook(A1 - CCSS Math)

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The vertex form of a quadratic function is given by f ( x) = a( x - h) 2 + k, where ( h, k) is the vertex of the parabola. FYI: Different textbooks have different interpretations of the reference " standard form" of a quadratic function.

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Let's just do one and you'll see how it works: See how these guys are the same, but with a different sign?

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Systems of equations word problems | Algebra 1 (practice) | Khan Academy

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Solve word problems by modeling them into a system of equations and solving it.

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This FREE lesson includes 3 pages of guided notes and a 1-page assignment. * This is part of my Linear Equations Unit Students learn about slope in the order

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About this unit

Use the power of algebra to understand and interpret points and lines (something we typically do in geometry). Topics you'll explore include the slope and the equation of a line.

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Graphing lines and slope | Algebra basics | Math | Khan Academy

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Use the power of algebra to understand and interpret points and lines (something we typically do in geometry). Topics you'll explore include the slope and the equation of a line.

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Varsity Tutors connects you to top tutors through its award-winning live learning platform for private in-home or online tutoring in your area.

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Graphing Systems of Linear Equations - Math Notes Algebra

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Systems of Linear Equations - Solve by Graphing CHANGE THE WAY your students see Algebra! These walkthrough worksheets can serve as your students notes, their homework, a reteaching resource, an enrichment resource, guided practice, assessment, and so much more!

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Sometimes we need to solve Inequalities like these:

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10
Solved!
How to Solve

Solving inequalities is very like solving equations, we do most of the same things ...

... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things can change the direction!

Safe Things To Do

These things do not affect the direction of the inequality:

Add (or subtract) a number from both sides

Multiply (or divide) both sides by a positive number

Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ("<" becomes ">" for example):

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x < 4

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

We went from this:

To this:

x+3 < 7

x < 4

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

7 < x

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:

Example: 3y < 15

If we divide both sides by 3 we get:

y < 5

And that is our solution: y < 5

Negative Values

When we multiply or divide by a negative number
we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from −3 to −7 is a decrease.

−7 < −3

7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality!

−2y < −8

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b, which gives us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.
So:
Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 < −10

Now add 3 to both sides:

x < −7

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example:

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

Now subtract 6 from each part:

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction.

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

Summary

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.

But these things will change direction of the inequality:

Multiplying or dividing both sides by a negative number

Swapping left and right hand sides

Don't multiply or divide by a variable (unless you know it is always positive or always negative)

447,448,303,304,449,450, 1222, 1223, 1224, 1225

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