# All About Standard Form

### Savreen Samra

## Learning Goals

- Understand how to find the zeros of a quadratic function.
- Know how to graph an equation that is in standard form by completing the square.
- Solve word problems involving completing the square or comlpeting the square using problem solving skills

## Summary

You can find all the specific things we learnt in more detail down in the Table of Contents.

## Table of Contents

- Completing The Square
- Quadratic Relations of the Form y = a (x - r) (x - s)
- Finding the X-intercept
- Quadratic Formula
- Discriminant 101
- All About Word Problems

## Completing The Square

y = ax² + bx + c → y = a(x - h)² + k

Rewrite **y = x² + 8x + 5** in the form **y = a(x - h)² + k **

## Step 1 Bracket the first two terms | ## Step 2 Take the middle coefficient (b) and divide by two. Then square that number. | ## Step 3 Add and subtract the number inside the bracket. |

## Step 4 Remove the last number from the bracket. Make sure that is there is a number in front of the bracket, you multiply it with that. | ## Step 5 Factor the perfect trinomial that is now inside the brackets. | ## Step 6 Simplify where needed |

For the equation above, you could be asked to sketch a graph using the vertex, axis of symmetry and two other points.

## Step 1 Find the vertex and axis of symmetry from the vertex form equation. | ## Step 2 Find the y-intercept by making x=0 in the standard form equation. | ## Step 3 Draw and Label these points on a graph, find the third point by reflecting the y-intercept off the axis of symmetry. |

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## Quadratic Relations of the Form

## y = a (x - r) (x - s)

Sketch the graph and label the x-intercepts, vertex and axis of symmetry for the quadratic relation** y = -4 (x-2) (x-4).**

**Determine an equation in the form y = a (x - r) (x - s)**

**The x-intercepts are (-7, 0) and (-3, 0).**

**The vertex is (-5, -2). **

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## Find the X-intercept

y = 2 (x -3)² - 32

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## Quadratic Formula

The quadratic formula us the x-intercepts of the quadratic equation.

Find the x-intercepts of the quadratic relation y = -5x² +8x -3 using the quadratic formula.

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## Discriminant 101

** b² - 4ac → Discriminant (D)**

This is the part of the quadratic formula that tells you how many x-intercepts/solutions the quadratic equation will have without having to use the whole formula.

## Zero SolutionsThis is when the value of (D) is less than zero. | ## One Solution This is when the value of (D) is zero. | ## Two Solutions This is when the value of (D) is more than zero. |

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## All About Word Problems

**Maximum and Minimum:**

a) Find the maximum or minimum point of the a parabola when given an equation in standard form.

- Change the standard form equation to vertex form.
- (k) is the maximum/minimum value.
- If (a) is positive, you are looking for the minimum value and if (a) is negative, you are looking for the maximum value.

**Path of a Ball (with completing the square):**

a) What is the maximum height of the object?

- Scroll up to see how to find the maximum height of a parabola.

b) At what horizontal distance does the object reach its maximum height?

- Change the standard form equation to vertex form.
- From there the value of (h) is the answer.
- make sure to remember to flip the sign of (h) when you remove it from the bracket.

* it is common that the "horizontal distance" could be replaced with time. *

**Maximum Revenue:**

First write the 'let' statements.

- "Let R be the total revenue, in dollars and x be the number of $____ decrease in price, in dollars."

Next write down two equations.

- Price equation → (regular price - the decrease of price x)
- Number of rentals equations → (the average rentals + the additional rentals x)

Next you multiply these two equations. Simplify until the equation is in standard form. From there you would complete the square and change the equation to vertex form.

From the vertex equation, take out (h) and substitute in into the price equation to get the price in which someone can maximize their revenue.

**Path of a Ball (with quadratic formula):**

a) How far has the ball traveled horizontally when it lands on the ground?

- Use the quadratic formula to find the x-intercepts.
- If you get more than 1 x-intercept, the one that is positive is the answer.

b) Find the horizontal distance when the ball is at a height of 4.5 m above the ground.

- Make h=4.5.
- First move the 4.5 to the other side and you will now have a trinomial on one side and 0 on the other,
- Find the x-intercepts using the quadratic formula.
- If you get two x-intercepts that is because the ball goes up then down, so it will reach 4.5 m twice.

* it is common that the "horizontal distance" could be replaced with time. *

**The Product of Two Consecutive Even Numbers:**

- "Let x be the first number and let x + 2 be the next number."

Next write the equation.

- x (x + 2)= product of the two numbers

Then move all the numbers to one side and use the quadratic formula to find the x-intercepts.

Finally, plug the x-intercepts into (x + 2) and you will get two sets of numbers, which are the answers.

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