# [zzz dead] unit 8 packet Quadratic Functions

This Unit 8 Packet contains 6 lessons and some additional practice:  
8-1 Exploring Quadratic Graphs  
8-2 Quadratic Functions (Part # 1)  
8-2 Quadratic Functions (Part # 2)  
Practice: 8-2 Quadratic Functions Worksheet  
8-3 Finding x - Intercepts of Quadratic Functions (Part # 1)  
Practice: 8-3 Finding x - Intercepts Worksheet #1  
8-4 Vertex Form and Transformations (Part 1)  
8-4 Vertex Form Worksheet # 1  
8-4 Vertex Form and Transformations (Part 2)  
8-4 Vertex Form Worksheet # 2

# Quadratics notes to understand wth is going on

# quadratics notes

<div class="flex flex-grow flex-col gap-3" id="bkmrk-quadratic-equation-s"><div class="min-h-[20px] flex flex-col items-start gap-4 whitespace-pre-wrap break-words"><div class="markdown prose w-full break-words dark:prose-invert light">- #### Quadratic equation standard form: ax^2 + bx + c = 0 This is the standard form for any quadratic equation, where a, b, and c are constants. It can be used to find the roots or zeros of the equation by using the quadratic formula.
- #### Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a This formula is used to find the roots (x-intercepts) of a quadratic equation in standard form. It gives the two possible values for x when the equation is equal to zero.
- #### Axis of symmetry: x = -b / 2a This formula is used to find the vertical line that divides the parabola into two symmetrical halves. It can help in graphing the quadratic function and finding the vertex.
- #### Vertex form: y = a(x - h)^2 + k This form is useful for easily identifying the vertex of a quadratic equation, where (h, k) is the vertex point. It can be used to graph the parabola or find the maximum/minimum value.
- #### Completing the square: x^2 + bx + c = a(x - h)^2 + k This method is used to convert a quadratic equation from standard form to vertex form. It involves adding and subtracting a constant term to make the left side a perfect square trinomial.
- #### Discriminant: Δ = b^2 - 4ac The discriminant is used to determine the nature of the roots of a quadratic equation. It can help predict whether the equation has real or complex roots and how many distinct solutions it has.
- #### Factored form: y = a(x - p)(x - q) This form is useful for quickly identifying the roots of a quadratic equation, where p and q are the x-intercepts. It can also be used to find the factors of the quadratic equation.
- #### Parabola focus: F(h, k + 1/4a) The focus is a point that defines the geometric property of a parabola. It can be used to find the directrix and derive the equation of a parabola from its geometric definition.
- #### Parabola directrix: y = k - 1/4a The directrix is a horizontal line that is equidistant from the focus and vertex of the parabola. It can be used in the geometric definition of a parabola and to find its equation from given points.

</div></div></div>

# The quadratic formula:

## x = (-b ± √(b^2 - 4ac)) / 2a

##### The formula gives you two possible results for the values of "x" that satisfy the quadratic equation y = ax^2 + bx + c.

##### The two possible values of "x" correspond to the x-coordinates of the points where the parabolic curve intersects the x-axis, also known as the x-intercepts or roots of the quadratic equation.

##### If the discriminant (b^2 - 4ac) is positive, then the quadratic equation has two real roots, and the parabolic curve intersects the x-axis at two distinct points. In this case, you will get two different values of "x" when you use the quadratic formula.

##### If the discriminant is zero, then the quadratic equation has one real root with a multiplicity of two, and the parabolic curve touches the x-axis at exactly one point. In this case, you will get the same value of "x" twice when you use the quadratic formula.

##### If the discriminant is negative, then the quadratic equation has two complex roots, and the parabolic curve does not intersect the x-axis. In this case, you will get two different complex values of "x" when you use the quadratic formula.

##### In summary, the two possible results from the quadratic formula correspond to the x-coordinates of the points where the parabolic curve intersects the x-axis, and the number and nature of these points depend on the value of the discriminant.

# how to find x and y intercepts

- Find the x-intercept by plugging in 0 for y.
- Find the y-intercept by plugging in 0 for x

# when i know the y value of the vertex, how do i find the x value?

If you know the y-coordinate of the vertex of a quadratic function, you can use the vertex form of the function to find the x-coordinate of the vertex.

The vertex form of a quadratic function is:

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

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

If you know the y-coordinate of the vertex, which is k in the vertex form equation, you can substitute it into the equation to get:

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

Simplifying this equation, we get:

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

Dividing both sides by "a", we get:

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

Taking the square root of both sides, we get:

√((y - k) / a) = x - h

Adding "h" to both sides, we get:

x = h ± √((y - k) / a)

So, to find the x-coordinate of the vertex, you can use the formula:

x = h ± √((y - k) / a)

where "h" is the x-coordinate of the vertex, "k" is the y-coordinate of the vertex, and "a" is the coefficient of the x^2 term in the quadratic function.

# 8-2 Quadratic Functions (Part # 1)

# Example 1

[![image.png](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/scaled-1680-/H575c2zPT1WIRQZS-image.png)](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/H575c2zPT1WIRQZS-image.png)

- A) The vertex is (4,3)
- B) The vertex is (-3,-3)

# Example 2 - Vertex Formula

##### [![image.png](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/scaled-1680-/Qlw65k44qSbInUWl-image.png)](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/Qlw65k44qSbInUWl-image.png)

##### a) y = 2x^2+4x 

1. 1. ##### we get the a, b, c in the formula <span style="text-decoration: underline;">y = ax^2 + bx + c</span>
        
        
        - ##### a = 2
        - ##### b = 4
        - ##### c = 0
    2. ##### then we plug in a, b into the formula
        
        
        - ##### x = -b / (2a)
        - ##### x = -4 / (2\*2) = -4/4 = -1

- ##### so the axis of symmetry is x = -1
- ##### to find the vertex
- ##### so the known vertex data is (-1, y)
- ##### to find y for the vertex we need to plug -1 into the formula y = 2x^2+4x
- ##### y = 2\*-1^2+4\*-1 = <span class="qv3Wpe" id="bkmrk--6">-6</span>
- ##### <span class="qv3Wpe">so the vertex is (-1, -6)</span>
    
    ##### <span class="qv3Wpe">-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=</span><span class="qv3Wpe">-=-=-=-=-=-=-=-=-=-=-=-=-=-=-</span>

##### B) y = -x^2 + 4x - 5   


1. 1. ##### we get the a, b, c in the formula <span style="text-decoration: underline;">y = ax^2 + bx + c</span>
        
        
        - ##### a = -1
        - ##### b = 4
        - ##### c = -5
    2. ##### then we plug in a, b into the formula
        
        
        - ##### x = -b / (2a)
        - ##### x = -4 / (2\*-1) = -4/-2 = <span class="qv3Wpe" id="bkmrk-2">2</span>

- ##### so the axis of symmetry is x = 2
- ##### to find the vertex
- ##### so the known vertex data is (2, y)
- ##### to find y for the vertex we need to plug -1 into the formula y = 2x^2+4x
- ##### y = -2^2 + 4 \* 2 - 5 = <span class="qv3Wpe" id="bkmrk--1">-1</span>
- ##### <span class="qv3Wpe">so the vertex is (2, <span class="qv3Wpe" id="bkmrk--1-1">-1</span>)</span>

# Example 3 + up/down test

[![image.png](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/scaled-1680-/7d9YkosVwD1B89qL-image.png)](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/7d9YkosVwD1B89qL-image.png)

##### up/down test

- opens upwards because a is Positive
- opens downward because a is Negative

##### Example 3

1. a) y= x^2 + 3x + 4 
    - opens upwards because a is Positive
2. b) y = -3x^2 + 5x  
    
    - opens downward because a is Negative
3. c) y = 2x - x^2 + 6  
    
    - opens downward because a is Negative

# Example 4 Graph f(x) = x^2 -2x - 8

#### Steps to Graph ax^2 + bx + c

- Find the vertex and the axis of symmetry. Sketch these in.
- Find the x-intercept by plugging in 0 for y.
- Find the y-intercept by plugging in 0 for x.
- Reflect your points across the axis of symmetry and connect your dots with a smooth U-shaped (not V-shaped) curve.

## Graph f(x) = x^2 -2x - 8 

[![image.png](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/scaled-1680-/rotl7yJ2OmLrywoV-image.png)](https://library.naruzkurai.tk/uploads/images/gallery/2023-04/rotl7yJ2OmLrywoV-image.png)

for now, I'm just gonna type my work and figure out what to do next

1. find the line of symmetry - 
    1. a = 1, b = -2, c = -8
2. use this to find the vertex 
    1. x = (b/2a)
    2. x = -(-2) / 2(1) = 1
3. since we know that the along the x axis at 1 will be the vertex we replace x with 1 in the original formula 
    1. x=1
    2. y = x^2 -2x - 8
    3. y = 1^2 + -2 \* 1 - 8 = 1- 2 - 8 = -9
    4. y = -9
4. the vertex is (1, -9)
5. since the vertex is -1,-9 we know that x=1 is the axis of symmetry
6. finding the y-intercept is the easiest to start with because we just replace x with 0
7. x = 0 | y = x^2 -2x - 8
8. y = 0 - 8 = -8
9. y-intercept = (0,-8)
10. so so To find the x-intercepts, you can set y equal to zero and solve for x:
11. y = 0 | x = (-b ± sqrt(b^2 - 4ac)) / 2a
    
    
    1. x = (-(-2) ± sqrt((-2)^2 - 4(1)(-8))) / 2(1)
        
        x = (2 ± sqrt(4 + 32)) / 2
        
        x = (2 ± sqrt(36)) / 2
        
        x = (2 ± 6) / 2
        
        x = 8 / 2 or x = -4/ 2
        
        x = 4 or x = -2
        
        sooooo (-2,0) &amp; (4,0)
12. so since we know 3 y axis points on the graph and the axis of symmetry we can get another point without doing much work   
    
    1. symmetry line = x = 1,
    2. calc'd x-intercept 0,-8 
        1. the symmetry line is 1 and the known point is 0 since 1-0 = 1 we can add that to the x coordinate of y and keep the same y coordinate to get the mirrored point making another point on the graph (2,-8)
    3. since we need one more point for the graph we can choose say x=3, | x^2 -2x - 8 
        1. y = 3^2 -3\*2 - 8 = <span class="qv3Wpe" id="bkmrk--5">-5</span>  
            
            1. soooo the new point is (3,-5) if we mirror that along 1,-9 we get (-1, -5 ) because 3 is 2 more than 1, and 2 less than 1 is -1. we also keep the same y coordinate
    4. so all points are: 
        1. (1, -9)
        2. (0,-8)
        3. (2, -8)
        4. (3,-5)
        5. (-1,-5)

# Example 5: Graph y = x^2 + 2x + 3

Find the vertex and the axis of symmetry. Sketch these in.  
• Find the x-intercept by plugging in 0 for y.  
• Find the y-intercept by plugging in 0 for x.  
• Reflect your points across the axis of symmetry and connect your dots with a smooth  
U-shaped (not V-shaped) curve.

fix the fallowing

- a = 1, b = 2, c = 3
- x^2 + 2x + 3

1. find the line of symmetry - 
    1. x = (b/2a)
    2. x = -(2) / 2(1) =1
2. use this to find the vertex
3. since we know that the along the x axis at -1 will be the vertex we replace x with 1 in the original formula 
    1. x=1
    2. y = x^2 -2x - 8
    3. y = 1^2 + -2 \* 1 - 8 = 1- 2 - 8 = -9
    4. y = -9
4. the vertex is (1, -9)
5. since the vertex is -1,-9 we know that x=1 is the axis of symmetry
6. finding the y-intercept is the easiest to start with because we just replace x with 0
7. x = 0 | y = x^2 -2x - 8
8. y = 0 - 8 = -8
9. y-intercept = (0,-8)
10. so so To find the x-intercepts, you can set y equal to zero and solve for x:
11. y = 0 | x = (-b ± sqrt(b^2 - 4ac)) / 2a
    
    
    1. x = (-(-2) ± sqrt((-2)^2 - 4(1)(-8))) / 2(1)
        
        x = (2 ± sqrt(4 + 32)) / 2
        
        x = (2 ± sqrt(36)) / 2
        
        x = (2 ± 6) / 2
        
        x = 8 / 2 or x = -4/ 2
        
        x = 4 or x = -2
        
        sooooo (-2,0) &amp; (4,0)
12. so since we know 3 y axis points on the graph and the axis of symmetry we can get another point without doing much work   
    
    1. symmetry line = x = 1,
    2. calc'd x-intercept 0,-8 
        1. the symmetry line is 1 and the known point is 0 since 1-0 = 1 we can add that to the x coordinate of y and keep the same y coordinate to get the mirrored point making another point on the graph (2,-8)
    3. since we need one more point for the graph we can choose say x=3, | x^2 -2x - 8 
        1. y = 3^2 -3\*2 - 8 = <span class="qv3Wpe" id="bkmrk--5">-5</span>  
            
            1. soooo the new point is (3,-5) if we mirror that along 1,-9 we get (-1, -5 ) because 3 is 2 more than 1, and 2 less than 1 is -1. we also keep the same y coordinate
    4. so all points are: 
        1. (1, -9)
        2. (0,-8)
        3. (2, -8)
        4. (3,-5)
        5. (-1,-5)

# Example 6: Graph y = 2x^2 - 8x

• Find the vertex and the axis of symmetry. Sketch these in.  
• Find the x-intercept by plugging in 0 for y.  
• Find the y-intercept by plugging in 0 for x.  
• Reflect your points across the axis of symmetry and connect your dots with a smooth  
U-shaped (not V-shaped) curve.

# Example 7: h = 16t^2 + 72t + 520

Suppose a particular “star” is projected from a firework at a starting height of 520 feet with an initial upward velocity of 72 ft/sec.

The equation:

h = 16t^2 + 72t + 520

gives the star’s height h in feet at time t in seconds.

  
a) How long will it take for the star to reach its maximum height?

b) What is the maximum height?

# quadratics vocabulary

# Quadratics Vocabulary

- Vertex 
    - The vertex is the highest or lowest point on the curve.
    - The vertex is the point (x, y) where x = -(b/2a).
    - Naru calls it the apex.
    - We then use this x-value in the equation to find the y-value of the vertex.
- Vertex Formula: 
    - The graph of y = ax^2 + bx + c has the line x = -(b/2a) as its axis of symmetry.
    - The x-coordinate of the vertex is x = -(b/2a)
    - You can find the y by plugging the x-cordonate into your equation.
- axis of symmetry 
    - the vertical line that splits the parabola down the middle.
    - The axis of symetry uses the same formula as the vertex formula to find the x-cordonate: x = -(b/2a)
- Up/ Down Test