sketching polynomials

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Sketching Polynomials John Du, Jen Tran & Thao Pham

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Sketching Polynomials. John Du, Jen Tran & Thao Pham. Things to know:. A polynomial function is a function of the form f(x) = a n xⁿ + a n-1 xⁿ¯¹ + … + a₁x + a₀ a n is the leading coefficient a₀ is the constant term n is the degree of the polynomial. - PowerPoint PPT Presentation

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Page 1: Sketching Polynomials

Sketching Polynomials

John Du, Jen Tran & Thao Pham

Page 2: Sketching Polynomials

Things to know:

• A polynomial function is a function of the form f(x) = anxⁿ + an-1 xⁿ¯¹ + … + a₁x + a₀

• a n is the leading coefficient • a₀ is the constant term• n is the degree of the polynomial

Page 3: Sketching Polynomials

• Here is a table of the common polynomials:

• Each graph curves differently depending on its degree.

Degree Type Standard Form0 Constant f(x) = a₀

1 Linear f(x) = a₁x + a₀

2 Quadratic f(x) = a₂x2 + a₁x + a₀

3 Cubic f(x) = a₃x3 + a₂x2 + a₁x + a₀

4 Quartic f(x) = a₄x⁴+a₃x3+a₂x2 +a₁x+a₀

Page 4: Sketching Polynomials

Linear • There are two forms of linear functions.• Slope-intercept Form: y = mx + b

- The slope of the graph is m.- The y-intercept of the graph is b.

• Standard Form: Ax + By = C-Find the x-intercept by making y = 0-Find the y-intercept by making x = 0

• The linear function crosses the x-intercept in a line.

Page 5: Sketching Polynomials

• Here’s an example:x – 4y = -8

• Make y = 0 to find the x-intercept, -8• Make x = 0 to find the y-intercept, 2

• Since a is positive, the graph is positive.

Page 6: Sketching Polynomials

Quadratics

• Quadratics have graphs that are “U” shaped, these are called parabolas.

• The x-intercepts of the parabola is called the zeros of the function.

• Quadratic function has the form y = ax² + bx + c

• The x-coordinate of the vertex is –(b/2a), which is also the axis of symmetry.

Page 7: Sketching Polynomials

• There are two forms of quadratic function:• Vertex Form: y = a(x – h)² + k

- The vertex is (h,k)- The axis of symmetry is x = h

• Intercept Form: y = a(x – p)(x – q)- The x-intercepts are p and q- The axis of symmetry is halfway

between (p,0) and (q,0)• If a > 0, then graph opens up. If a < 0,

graph opens down.

Page 8: Sketching Polynomials

• Let’s look at an example:y = (x + 2)(x – 3)

• This equation is in intercept form so you should graph the zeros first.

Page 9: Sketching Polynomials

• Since a is 1, we know that the graph will open up.

• We know that the vertex is between (p,0),(q,0) so you plot the vertex and draw the graph.

Page 10: Sketching Polynomials

Cubic

• Basically you now know how to graph polynomials. It is important to graph the intercepts on the graphs.

• Then, you have to think about the end behavior of the graph.

• For a cubic function, the graph looks like an “S” or a squiggly line.

• Ex: or

Page 11: Sketching Polynomials

• Cubic functions look like:f(x)= ax + bx + cx + d

• The parent function of cubic functions is simply: f(x)=x

Page 12: Sketching Polynomials

f(x) = ax + bx + cx + d• Coefficient "a" in the equation above is to make the

graph "wider" or "skinnier", or to reflect it (if negative.)• Coefficient "b" represents a quadratic function.• Coefficient "c" represents a linear function.• Coefficient "d " is the y-intercept.

(In this case, a=1, therefore all 3 lines are

neither skinnier or wider than the other.)

Page 13: Sketching Polynomials

There are 2 different ways to determine whether a graph moves/left/right.• If f(x) = (x + a) and a > 0, the graph undergoes a horizontal shift d units to the left.

• If f(x) = (x + a) and a < 0, the graph undergoes a horizontal shift d units to the right.

The equation for this is

f(x) = (x+1)

Because 1 > 0,

the graph moves to the left 1 space.

Page 14: Sketching Polynomials

Quartic

• A quartic graph looks like a “W” or a "M"• Ex: or

Page 15: Sketching Polynomials

• The equation of a quartic looks like this model: f(x)= ax + bx +cx +dx+ e

• The parent function of a quartic is f(x)= x• The leading coefficient determines of the

graph faces up or down.

Page 16: Sketching Polynomials

• If a is negative, both side of the graph will go to negative infinity.

• If a is positive, both side of the graph will go to positive infinity.

Page 17: Sketching Polynomials

Understanding

Knowing the the rules about powers, it helps you to predict what the graphs would look like without graphing it all.

Usually you would first graph the zeros. The importance of zeros are that they're the x-intercepts and it is the locations where the changes of the graph happens.

Page 18: Sketching Polynomials

Applying the skills

f(x) = (x-5)³ (x+4)² (x-1)• First, identify the zeros: 5, -4, and 1• Then, predict the curve of the graph

at each zero: at x = 5, the graph curves through, at x = -4, that is the vertex of a parabola and at x = 1, the graph just goes through because it is linear.

Page 19: Sketching Polynomials

Here's what the graph should look like!

Page 20: Sketching Polynomials

For more practice

Algebra II Textbook:• Page 334 Questions 49 - 52• Page 335 Questions 65 - 79

Page 21: Sketching Polynomials

For more help!!

Here is a useful video!