Quadratic Functions and Transformations Study Notes

Quadratic Functions and Transformations Study Guide
Graphing Quadratic Functions

Quadratic functions graph as parabolas. The general form of a basic quadratic function is y=ax2y=ax2.

  • Key characteristics of y=ax2y=ax2:

    • If a>0a>0, the parabola opens upwards.

    • If a<0a<0, the parabola opens downwards.

    • If ∣a∣>1∣a∣>1, the graph is vertically stretched (appears narrower) compared to y=x2y=x2.

    • If 0<∣a∣<10<∣a∣<1, the graph is vertically compressed (appears wider) compared to y=x2y=x2.

    • The vertex is at (0,0)(0,0) and the axis of symmetry is x=0x=0.

  • Examples to visualize:

    1. y=3x2y=3x2: Opens upwards, stretched vertically.

    2. f(x)=−5x2f(x)=−5x2: Opens downwards, stretched vertically.

    3. y=x2y=x2: Parent function, opens upwards.

    4. f(x)=−5x2f(x)=−5x2: (Repeat of 2)

    5. f(x)=x2f(x)=x2: (Repeat of 3)

    6. f(x)=x2+4f(x)=x2+4: This function is a vertical translation of f(x)=x2f(x)=x2.

Translations of the Parent Function f(x)=x2f(x)=x2

Translations shift the parabola without changing its shape or orientation. The parent function is f(x)=x2f(x)=x2.

  • Identified Translations:

    1. Function: f(x)=x2+4f(x)=x2+4

      • Transformation: Translated 4 units up. (Graph shifts vertically).

    2. Function: f(x)=(x−3)2f(x)=(x−3)2

      • Transformation: Translated 3 units to the right. (Graph shifts horizontally).

Key Features of Quadratic Functions in Vertex Form

For a quadratic function in vertex form, y=a(x−h)2+ky=a(x−h)2+k, specific features can be easily identified.

  • Features to identify:

    1. Function: y=(x−2)2+3y=(x−2)2+3

      • Vertex: (h,k)=(2,3)(h,k)=(2,3)

      • Axis of Symmetry: x=h=2x=h=2

      • Minimum Value: k=3k=3 (Since a=1>0a=1>0, the parabola opens upwards)

      • Domain: All real numbers (often denoted as (−ho,ho)(−ho,ho)).

      • Range: All real numbers greater than or equal to kk: yho3yho3

    2. Function: f(x)=−0.2(x+3)2+2f(x)=−0.2(x+3)2+2

      • Vertex: (h,k)=(−3,2)(h,k)=(−3,2)

      • Axis of Symmetry: x=h=−3x=h=−3

      • Maximum Value: k=2k=2 (Since a=−0.2<0a=−0.2<0, the parabola opens downwards)

      • Domain: All real numbers ((−ho,ho)(−ho,ho)).

      • Range: All real numbers less than or equal to kk: yho2yho2

Additional Graphs and Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror images. For vertex form y=a(x−h)2+ky=a(x−h)2+k, the axis of symmetry is x=hx=h.

  • Examples:

    1. Function: y=(x+2)2−1y=(x+2)2−1

      • Axis of Symmetry: x=−2x=−2

    2. Function: y=−4(x−3)2+2y=−4(x−3)2+2

      • Axis of Symmetry: x=3x=3

Modeling Quadratic Functions from Graphs

Given the vertex (h,k)(h,k) and another point (x,y)(x,y) from a graph, you can write the quadratic function in vertex form y=a(x−h)2+ky=a(x−h)2+k by solving for aa.

  • Examples:

    1. If a graph has a vertex at (2,3)(2,3) and passes through another point:

      • Resulting Function: y=(x−2)2+3y=(x−2)2+3

    2. If a graph has a vertex at (−3,−1)(−3,−1) and passes through another point (e.g., (−2,1)(−2,1), then 1=a(−2−(−3))2−1ho1=a(1)2−1ho2=a1=a(−2−(−3))2−1ho1=a(1)2−1ho2=a):

      • Resulting Function: y=2(x+3)2−1y=2(x+3)2−1

Transformation of Quadratic Functions from Parent Function (y=x2y=x2)

The general vertex form y=a(x−h)2+ky=a(x−h)2+k shows all transformations:

  • aa: Vertical stretch/compression (if ∣a∣>1∣a∣>1 or 0<∣a∣<10<∣a∣<1) and reflection across the x-axis (if a<0a<0).

  • hh: Horizontal translation (right if x−hx−h, left if x+hx+h).

  • kk: Vertical translation (up if +k+k, down if −k−k).

  • Transformation examples:

    1. Function: y=3(x+2)2y=3(x+2)2

      • Transformations: Translate 2 units left; stretch vertically by a factor of 3.

    2. Function: y=−(x+5)2+1y=−(x+5)2+1

      • Transformations: Translate 5 units left; reflect across the x-axis; translate 1 unit up.

    3. Function: y = rac{3}{2}(x + 4)^2 - 2

      • Transformations: Translate 4 units to left; stretch vertically by factor rac32rac32 (correction from original note); translate 2 units down.

    4. Function: y=−0.08(x−0.04)2+1.2y=−0.08(x−0.04)2+1.2

      • Transformations: Translate 0.04 units to right; compress vertically by factor 0.08; reflect across x-axis; translate 1.2 units up.

Writing Quadratic Equations in Vertex Form

To write a quadratic equation in vertex form y=a(x−h)2+ky=a(x−h)2+k given a vertex (h,k)(h,k) and another point (x,y)(x,y), substitute the vertex coordinates into the formula. Then, substitute the point's coordinates for xx and yy and solve for aa.

  • Examples:

    1. Given Vertex: (3,−2)(3,−2), Point: (2,3)(2,3)

      • Substitute: 3=a(2−3)2−2ho3=a(−1)2−2ho3=a−2hoa=53=a(2−3)2−2ho3=a(−1)2−2ho3=a−2hoa=5

      • Equation: y=5(x−3)2−2y=5(x−3)2−2

    2. Given Vertex: (1,1)(1,1), Point: (2,−8)(2,−8)

      • Substitute: −8=a(2−1)2+1ho−8=a(1)2+1ho−8=a+1hoa=−9−8=a(2−1)2+1ho−8=a(1)2+1ho−8=a+1hoa=−9

      • Equation: y=−9(x−1)2+1y=−9(x−1)2+1 (Correction from original point in note)

    3. Given Vertex: (−4,−24)(−4,−24), Point: (−5,−25)(−5,−25)

      • Substitute: −25=a(−5−(−4))2−24ho−25=a(−1)2−24ho−25=a−24hoa=−1−25=a(−5−(−4))2−24ho−25=a(−1)2−24ho−25=a−24hoa=−1

      • Equation: y=−(x+4)2−24y=−(x+4)2−24

    4. Given Vertex: (−12.5,35.5)(−12.5,35.5), Point: (1,400)(1,400)

      • Substitute: 400=a(1−(−12.5))2+35.5ho400=a(13.5)2+35.5ho400=182.25a+35.5ho364.5=182.25ahoa=2400=a(1−(−12.5))2+35.5ho400=a(13.5)2+35.5ho400=182.25a+35.5ho364.5=182.25ahoa=2

      • Equation: y=2(x+12.5)2+35.5y=2(x+12.5)2+35.5

Applications of Quadratics

Quadratic functions are used to model real-world scenarios, particularly those involving maximum or minimum values (e.g., projectile motion, area optimization). The vertex represents these maximum or minimum points.

  • Examples:

    1. Curtains Area Function: A=−4x2+40xA=−4x2+40x

      • Vertex (x-coordinate): x = - rac{b}{2a} = - rac{40}{2(-4)} = - rac{40}{-8} = 5 ft (This is the width for max area)

      • Maximum Area Calculation: A=−4(5)2+40(5)=−4(25)+200=−100+200=100A=−4(5)2+40(5)=−4(25)+200=−100+200=100 ft²

    2. Rocket Height Model: f(x)=−1.5(x−16)2+384f(x)=−1.5(x−16)2+384

      • This model describes the height of a rocket over time. The vertex (16,384)(16,384) indicates the rocket reaches a maximum height of 384 units at 16 units of time.

    3. Enclosure Area Function: A=−2x2+120xA=−2x2+120x

      • Maximized Area Dimensions: For a fence of length 120120, if one side is xx (and another xx), then the third side is 120−2x120−2x. Area A=x(120−2x)=−2x2+120xA=x(120−2x)=−2x2+120x.

      • Vertex (x-coordinate for max area): x = - rac{b}{2a} = - rac{120}{2(-2)} = - rac{120}{-4} = 30 ft.

      • If x=30x=30 ft, the other sides are 6060 ft. So, dimensions are 30extftimes60extft30extftimes60extft.

Standard Form of a Quadratic Function

The standard form of a quadratic function is y=ax2+bx+cy=ax2+bx+c. To find the vertex from standard form, calculate the x-coordinate of the vertex using x = - rac{b}{2a}, then substitute this value back into the function to find the y-coordinate.

  • Vertex and Range Identification:

    1. Function: y=x2−4x+1y=x2−4x+1

      • Vertex: x = - rac{-4}{2(1)} = 2. y=(2)2−4(2)+1=4−8+1=−3y=(2)2−4(2)+1=4−8+1=−3. So, vertex is (2,−3)(2,−3).

      • Axis of Symmetry: x=2x=2

      • Range: Since a=1>0a=1>0 (opens upwards), yho−3yho−3

    2. Function: y=x2+2x+3y=x2+2x+3

      • Vertex: x = - rac{2}{2(1)} = -1. y=(−1)2+2(−1)+3=1−2+3=2y=(−1)2+2(−1)+3=1−2+3=2. So, vertex is (−1,2)(−1,2).

      • Minimum: (−1,2)(−1,2) (since a=1>0a=1>0, it's a minimum)

      • Range: Since a=1>0a=1>0 (opens upwards), yho2yho2

    3. Function: y=2x2+3x−5y=2x2+3x−5

      • Vertex: x = - rac{3}{2(2)} = - rac{3}{4}. y = 2(- rac{3}{4})^2 + 3(- rac{3}{4}) - 5 = 2( rac{9}{16}) - rac{9}{4} - 5 = rac{9}{8} - rac{18}{8} - rac{40}{8} = - rac{49}{8}. So, vertex is (- rac{3}{4}, - rac{49}{8}).

      • Minimum: Value at (- rac{3}{4}, - rac{49}{8}) (since a=2>0a=2>0, it's a minimum)

      • Axis of Symmetry: x = - rac{3}{4}

      • Range: Since a=2>0a=2>0 (opens upwards), y
        ho - rac{49}{8}

Solving Quadratic Equations through Various Methods

To find the x-intercepts (roots/zeros) of a quadratic function, set y=0y=0 and solve for xx. There are several methods.

Factoring Quadratic Expressions

Factoring involves breaking down a quadratic expression into a product of linear expressions (e.g., (x+p)(x+q)(x+p)(x+q)). This method is efficient when the quadratic is easily factorable.

  • Factoring Examples:

    1. Expression: x2+11x+28x2+11x+28

      • Factored Form: (x+7)(x+4)(x+7)(x+4)

    2. Expression: 5x2+25x−705x2+25x−70

      • Factored Form: 5(x2+5x−14)=5(x−2)(x+7)5(x2+5x−14)=5(x−2)(x+7)

    3. Expression: 6x2+16x2+1

      • Factored Form: This expression is generally not factorable over real numbers, as it represents a parabola that opens upwards with its vertex above the x-axis, so it has no real roots. (Correction from original note)

Completing the Square and Quadratic Formula

  • Completing the Square: This method transforms a quadratic equation into the form (xhoh)2=d(xhoh)2=d, from which solutions can be found by taking the square root of both sides. It's also used to convert standard form to vertex form.

    1. Example (Solving by Completing the Square): x2+12x+36−25=0x2+12x+36−25=0

      • This simplifies to (x+6)2−25=0(x+6)2−25=0

      • (x+6)2=25(x+6)2=25

      • x+6=ho5x+6=ho5

      • x=−6ho5x=−6ho5

      • Resulting Solutions: x=−1x=−1 or x=−11x=−11

  • The Quadratic Formula: A universal method for solving any quadratic equation ax2+bx+c=0ax2+bx+c=0.

    • Formula: x = rac{-b
      ho
      ho{b^2 - 4ac}}{2a}

The Quadratic Formula Applications

Use the quadratic formula to find solutions (roots) to any quadratic equation.

  • Finding Solutions:

    1. Function: x2−8x+15=0x2−8x+15=0

      • Here, a=1,b=−8,c=15a=1,b=−8,c=15

      • x = rac{-(-8)
        ho
        ho{(-8)^2 - 4(1)(15)}}{2(1)}

      • x = rac{8
        ho
        ho{64 - 60}}{2}

      • x = rac{8
        ho
        ho{4}}{2}

      • x = rac{8
        ho 2}{2}

      • Roots: x = rac{8+2}{2} = 5 and x = rac{8-2}{2} = 3

    2. Tide Height Function: h = rac{1}{2} t^2 + 6t - 9

      • This function models the height of the tide over time. To find when the tide reaches a certain height, set hh to that value and solve the resulting quadratic equation using the quadratic formula.

Discriminant Applications

The discriminant is the part of the quadratic formula under the square root, ho=b2−4acho=b2−4ac. It indicates the number and type of solutions (roots) a quadratic equation has without actually solving it.

  • Discriminant Evaluation:

    1. Condition: x2+5x+8=0x2+5x+8=0

      • Here, a=1,b=5,c=8a=1,b=5,c=8

      • Value: ho=(5)2−4(1)(8)=25−32=−7ho=(5)2−4(1)(8)=25−32=−7

      • Solutions: Since ho<0ho<0, there are no real solutions (two complex solutions).

    2. Practical Settings: The discriminant can be used to determine if a projectile will reach a certain height (if hoho0hoho0 for height equation) or if a cost function will ever break even (hoho0hoho0 for profit equation).

Summary Note

This study guide covers quadratic functions, their transformations, different forms (vertex and standard), methods for graphing, and techniques for solving quadratic equations (factoring, completing the square, quadratic formula). Understanding the vertex, axis of symmetry, and discriminant is crucial for analyzing and applying quadratic functions in various contexts. Pay attention to the role of the 'a' coefficient in determining the shape and direction of the parabola, and the 'h' and 'k' values in determining its position.