Sketching Polynomials

John Du, Jen Tran & Thao Pham

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

• 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

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.

• 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.

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.

• 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.

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

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

• 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.

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

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

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

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.)

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.

Quartic

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

• 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.

• 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.

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.

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.

Here's what the graph should look like!

For more practice

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

For more help!!

Here is a useful video!