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MHF4U Lesson Notes
Polynomial Equations and Inequalities
2.5 Polynomial Functions: Polynomial Equations
General Shape of Graphs:
x2 x3
Upwards Quadrant 1 to Quadrant 3
Sketching Graphs
If the equation is given, we can find other factors that will contribute to the sketch of the graph
1. Point where the curve intersect the y-axis
a. Substituting all x values of the equation with 0 (x = 0)
2. Point where the curve intersect the x-axis
a. Substituting all y values of the equation with 0 (y = 0)
3. Turning points: The gradient of the graph that switches between positive and negative values)
a. Maximum or Minimum value
b. TIP: Number of turning points IS ALWAYS one value decreased of the degree value
4. End Behaviors *varies on the degree of parent function and the value of a (negative/positive)
a. The value of y when x is either (a) very large and positive or (b) very large and negative
b.
+x2: (x → ±∞ , y → ∞) +x3: (x → -∞ , y → -∞)
(x → +∞ , y → +∞)
-x2: (x → ±∞ , y → -∞) -x3: (x → -∞ , y → +∞)
(x → +∞ , y → -∞)
Graphs of Cubic Functions
1. Standard Form: f(x) = ax3 + bx2 + cx + d
a. Characteristics of the graph depends on the values of coefficients
, 2
i.
2. Factored Form: f(x) = a(x - p)(x - q)(x - r)
a. Intersects the x axis at the points (p, 0)(q, 0)(r, 0)
i. p, q, and r are the roots to the cubic function
b. To sketch a graph in factored form:
i. Find the roots then plot it on the x axis
ii. Find the y intercept (substitute 0 into x; x = 0)
iii. Look at the coefficient of x3 to determine whether it’s positive or negative
(indicating direction of end behavior)
Using Graphs to Solve Equation
- In other words, it is to find the Point of Intersection. These methods can be applied to any parent
function (x2 or x3). x3 has 3 turning points (not necessarily roots) while x2 has 2 turning points
SAMPLE EQUATION: 4x2 - 10 = 6x
Method 1: Consider the left side of the equation and the right side as two separate functions
LS RS
f(x) = 4x2 - 10 f(x) = 6x
The points where these two intersect will give us the solution to
the equation.
The graph of (y = 4x2 - 10) and (y = 6x) intersect at points (2.5,
15) and (-1, -6). The x values of these coordinates give us the
solutions to the equation 4x2 - 10 = 6x as: (x = -1) and (x = 2.5)
MHF4U Lesson Notes
Polynomial Equations and Inequalities
2.5 Polynomial Functions: Polynomial Equations
General Shape of Graphs:
x2 x3
Upwards Quadrant 1 to Quadrant 3
Sketching Graphs
If the equation is given, we can find other factors that will contribute to the sketch of the graph
1. Point where the curve intersect the y-axis
a. Substituting all x values of the equation with 0 (x = 0)
2. Point where the curve intersect the x-axis
a. Substituting all y values of the equation with 0 (y = 0)
3. Turning points: The gradient of the graph that switches between positive and negative values)
a. Maximum or Minimum value
b. TIP: Number of turning points IS ALWAYS one value decreased of the degree value
4. End Behaviors *varies on the degree of parent function and the value of a (negative/positive)
a. The value of y when x is either (a) very large and positive or (b) very large and negative
b.
+x2: (x → ±∞ , y → ∞) +x3: (x → -∞ , y → -∞)
(x → +∞ , y → +∞)
-x2: (x → ±∞ , y → -∞) -x3: (x → -∞ , y → +∞)
(x → +∞ , y → -∞)
Graphs of Cubic Functions
1. Standard Form: f(x) = ax3 + bx2 + cx + d
a. Characteristics of the graph depends on the values of coefficients
, 2
i.
2. Factored Form: f(x) = a(x - p)(x - q)(x - r)
a. Intersects the x axis at the points (p, 0)(q, 0)(r, 0)
i. p, q, and r are the roots to the cubic function
b. To sketch a graph in factored form:
i. Find the roots then plot it on the x axis
ii. Find the y intercept (substitute 0 into x; x = 0)
iii. Look at the coefficient of x3 to determine whether it’s positive or negative
(indicating direction of end behavior)
Using Graphs to Solve Equation
- In other words, it is to find the Point of Intersection. These methods can be applied to any parent
function (x2 or x3). x3 has 3 turning points (not necessarily roots) while x2 has 2 turning points
SAMPLE EQUATION: 4x2 - 10 = 6x
Method 1: Consider the left side of the equation and the right side as two separate functions
LS RS
f(x) = 4x2 - 10 f(x) = 6x
The points where these two intersect will give us the solution to
the equation.
The graph of (y = 4x2 - 10) and (y = 6x) intersect at points (2.5,
15) and (-1, -6). The x values of these coordinates give us the
solutions to the equation 4x2 - 10 = 6x as: (x = -1) and (x = 2.5)
