Quadratic - Intermediate Algebra - Lecture Slides, Slides of Algebra

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§8.3 Quadratic

Fcn Graphs

Review §

 Any QUESTIONS About

• §8.2 → Quadratic Eqn Applications

 Any QUESTIONS About HomeWork

• §8.2 → HW-

8.2 MTH 55

Making Complete Plots

1. Arrows in POSITIVE **Direction Only

2. Label x & y axes** on POSITIVE **ends
3. Mark and label at** least one unit on **each axis
4. Use a ruler for** Axes & **Straight-Lines
5. Label significant** points or quantities

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 

^  

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 

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Graphs of Quadratic Eqns

• All quadratic functions have graphs similar to y = x^2. Such curves are called parabolas. They are U-shaped and symmetric with respect to a vertical line known as the parabola’s line of symmetry or axis of symmetry.
• For the graph of f ( x ) = x^2 , the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.

Example  Graph f ( x ) = − 3 x^2

• Solution: Make T-Table and Connect-Dots

 Same Axis & Vertex but opens DOWNward

x y ( x, y )

0 1

• 2

0

(0, 0) (1, –3) (–1, –3) (2, –12) (–2, –12)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

2

3

1

6 45

Examples of ax^2 Parabolas

f ( ) x = x^2

4 3

6

2

5

1

f ( ) x = 4 x^2

( ) 1 2 4

f x = x

f ( ) x = − x^2

x

y

The Graph of f ( x ) = a ( x – h )^2

• We could next consider graphs of f ( x ) = ax^2 + bx + c , where b and c are not both 0.
• It turns out to be more convenient to first graph f ( x ) = a ( x – h )^2 , where h is some constant; i.e., h is a NUMBER
• This allows us to observe similarities to the graphs drawn in previous slides.

Example  Graph f ( x ) = ( x −2) 2

• Solution: Make T-Table and Connect-Dots

 The Vertex SHIFTED 2-Units to the Right

x y ( x, y ) 0 1

• 2 3 4

4 1 9 0 1 4 (0, 4) (1, 1) (–1, 9) (2, 0) (3, 1) (4, 4)

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

4 3

6

2

5

1

78

vertex

Graph of f ( x ) = a ( x – h ) 2 + k

• Given a graph of f ( x ) = a ( x – h)^2 , what happens if we add a constant k?
• Suppose we add k = 3. This increases f ( x ) by 3, so the curve moves up - If k is negative , the curve moves down.
• The axis of symmetry for the parabola remains x = h , but the vertex will be at ( h , k ), or equivalently ( h , f ( h )) - f ( h ) = a ([ h ] – h ) 2 + k = 0 + k → f ( h ) = k

Example  Graph

 The Vertex SHIFTED 3-Units Left and 1-Unit Down

 Make T-Table and Connect-Dots x y ( x, y ) 0

-11/

–3/

–3/

(0, -11/2) (–1, –3) (–2, –3/2) (–3, –1) (–4, –3/2) (–5, –3)

( ) 1 ( 3) 2 1. 2

f x = − x + −

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

2

3

1

-5 -

vertex^ -

Graphing y = f ( x ) = a ( x – h )^2 + k

1. The graph is a parabola. - Identify a , h , and k 2. Determine how the parabola opens. - If a > 0 (positive), the parabola opens up. - If a < 0 (negative), the parabola opens down. 3. Find the vertex. The vertex is ( h , k ). - If a > 0 (or a < 0 ), the function f has a minimum (or a maximum ) value k at x = h

Graphing y = f ( x ) = a ( x – h )^2 + k

4. Find the x-intercepts. - Find the x-intercepts (if any) by setting f ( x ) = 0 and solving the equation a ( x – h ) 2 + k = 0 for x.

• Solve by: AdditionPrin + MultPrin + SqRtPrin + AdditionPrin
• If the solutions are real numbers, they are the x- intercepts.
• If the solutions are NOT Real Numbers, the parabola either lies above the x–axis (when a > 0) or below the x–axis (when a < 0).

Example  Graph f^ ( ) x =^2 ( x^^ −^3 )

 SOLUTION

Step 1 a = 2, h = 3, and k = – Step 2 a = 2, a > 0, the parabola opens up. Step 3 ( h , k ) = (3, –8); the function f has a minimum value –8 at x = 3. Step 4 Set f ( x ) = 0 and solve for x.

0 = (^2) ( x − (^3) )^2 − 8 8 = (^2) ( x − (^3) )^2 4 = (^) ( x − (^3) )^2

x − 3 = ± 2 x = 5 or x = 1 x -intercepts: 1 and 5

Example  Graph f^ ( ) x =^2 ( x^^ −^3 )

 SOLUTION cont. Step 5 Replace x with 0.

f (^) ( ) 0 = 2 0( − (^3) ) 2 − 8 = 2 9( ) − 8 = 10 y -intercept is 10.

Step 6 axis: x = 3, opens up, vertex: (3, –8), passes through (1, 0), (5, 0) and (0, 10), the graph is y = 2 x^2 shifted three units right and eight units down.