Intro to and Solving Quadratics

Quadratic Functions

By Liana C

Multiple Choice

Which parent function matches the graph?

• y = x

• y = x²

• y = x³

• y = 2^x

Multiple Choice

Which solid parabola, h, represents the equation $$y=x^2+2$$ ?

The intercepts and the vertex are the points you need to fully graph a quadratic.

Each of these points are solved with one of the 3 forms of a quadratic.

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Special Points

Multiple Choice

Which form of a quadratic function, when solved, will give you the x-intercepts?

• Standard Form

  $$y=ax^2+bx+c$$

• Factored Form

  $$y=a\left(x-r_1\right)\left(x-r_2\right)$$

• Vertex Form

  $$y=a\left(x-h\right)^2+k$$

Multiple Choice

What are the x-intercepts of this quadratic?

• (2, 0) and

  (4, 0)

• (-2, 0) and (-5, 0)

• (2, 0) and

  (5, 0)

• (-2, 0) and

  (-4, 0)

Multiple Choice

What is the y-intercept of this parabola?

• (−1, 3)
• (−3, 11)
• (−2, 5)
• (0, 5)

Multiple Choice

What is the vertex of this quadratic?

• Maximum;

  (-4, 2)

• Minimum;

  (-4, 2)

• Maximum;

  (2, -4)

• Minimum;

  (2, -4)

Multiple Choice

What is the equation for this parabola in factored form?

• y=(x+2)(x-1)
• y=(x-2)(x+1)
• y=(x-2)(x-1)
• y=(x+2)(x+1)

Parabolas mimic a common phenomena: a falling object over time, t.

The function here shows the height of a ball thrown. The intercepts and vertex all have special meanings when paired with a real context.

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​Quadratics in the world

Vertex: (t, h) a maximum or minimum height of the object at a specific time.

​Y-intercept: (0, h) the starting height of the object at time 0.

X-intercept: (t, 0) the time when the object is at a height of 0.

​In Context using feet/time

Multiple Choice

The graph below shows the height, in feet, over time, in seconds, of a student's rocket launch in science class.

How long was the rocket in the air?

• 14 minutes

• 7 seconds

• 14 seconds

• 7 minutes

Multiple Choice

The graph below shows the height, in feet, over time, in seconds, of a student's rocket launch in science class.

What was the maximum height the rocket reached?

• 25 feet

• 45 feet

• 65 feet

• 85 feet

Multiple Choice

The graph below shows the height, in feet, over time, in seconds, of a student's rocket launch in science class.

What would the equation be of the function in factored form?

• $$y=-a\left(x+1\right)\left(x+14\right)$$

• $$y=-ax\left(x-14\right)$$

• $$y=-a\left(x+0\right)\left(x+14\right)$$

• $$y=a\left(x\right)\left(x-14\right)$$

-Set equation equal to zero.
-Factor.
-Set each factor equal to zero.

Zero Product Property

-Type the equation in the calc.
-See points in the table where y = 0.
-If not whole numbers, use [CALC] [ZERO].

Graphing

Quadratic Formula

​Solving Quadratics: finding x-intercepts

Solve y = 2x² - 4x + 6

By Factoring

1. Set y = 0.

2. Look for a GCF.

3. Factor the trinomial.

4. Use the __________________________ _______________________________________ Property:

   ("If a product of factors equals zero, then

   one of those factors must be zero"). Set

   factor equal to zero and solve.

Multiple Choice

Factor
$$x^2+12x+20$$  

• $$\left(x+2\right)\left(x+10\right)$$  

• $$\left(x+4\right)\left(x+5\right)$$  

• $$\left(x-2\right)\left(x-10\right)$$  

• $$\left(x-4\right)\left(x-5\right)$$  

Multiple Choice

Solve: x² - 9x + 20 = 0

• x = -4 and x = -5

• x = 20 and x = -9

• x = 4 and x = 5

• x = -4 and x = 5

Multiple Choice

Use the Zero Product Property to solve for n.

Multiple Choice

Solve using the Zero-Product Property

(x −11)(5x + 1) = 0

• x = 11 or x = $$-\frac{1}{5}$$

• x = −11 or x = $$-\frac{1}{5}$$

• x = 11 or x = $$\frac{1}{5}$$

• x = −11 or x = $$\frac{1}{5}$$

Multiple Choice

Solve:

m² − 100 = 0

• m = 50 , −50

• m = 10

• m = 10, −10

• m = −10

Solve y = 2x² - 4x + 6

By Graphing

Press [Y = ]: Type equation into Y₁.
Option 1
→[GRAPH]: Estimate the intercepts. Can use [TRACE] with these values.
→[2nd][GRAPH]: Find "_____________" value(s) for when "_____________" = 0.

Option 2
→[CALC][ZERO]: Cursor over of the zeros.
→Cursor "__________________________" of intercept, [ENTER]. Cursor "__________________________" of intercept, [ENTER].
→Do not type anything for "Guess?"; press [ENTER].

Multiple Choice

Use the graph to determine the solutions.

• -1 and -3
• 1 and -3
• 1 and 3
• -1 and 3

Multiple Choice

Take an educated guess!

If the graph of a quadratic does not intercept the x axis at any point, then it has:

• 1 Real Solution

• 2 Real Solutions

• 1 Imaginary Solution

• No Real Solution

Solve y = 2x² - 4x + 6

By the Quadratic Formula

Multiple Choice

In the equation
y = x² +5x +7, match each leading coefficient with its correct letter 

• a=0, b=5, c=7
• a=1, b=5, c=7
• a=7, b=5, c=1

Multiple Choice

The quadratic equation can be used to solve quadratic equations that cannot be factored.

• True
• False

Multiple Choice

Select the false statement about the parabola

• The vertex is at (-2,-3)

• The minimum value is -3

• There are no real solutions.

• The y-intercept is

  (0, -7).