This guide summarizes the core concepts you've used across the problems, with each idea illustrated through specific examples.
1. Finding Intersection Points of Two Functions
Key idea: Intersection points occur where the outputs of two functions are equal.
Steps:
- Set the functions equal: Solve $f(x) = g(x)$.
- Move everything to one side so the equation equals 0.
- Factor if possible—factored form almost always makes the problem easier.
- Solve each factor by setting it equal to zero.
- Plug x-values into either function to get full coordinate points.
Example 1:
Functions: $f(x) = x^2(x+1)$, $g(x) = x+1$
- Set equal: $x^2(x+1) = x+1$
- Factor: $(x+1)(x^2-1) = 0 \to (x+1)^2(x-1) = 0$
- Solutions: $x=-1$, $x=1$
- Plug in: Points are $(-1, 0)$ and $(1, 2)$.
Example 2:
Functions: $(x+5)(x-2)$ and $(2x+1)(x-2)$
- Shared factor $(x-2)$ → gives intersection at $(2, 0)$.
- Cancel shared factor, solve remaining equation.
- Second intersection at $(4, 18)$.
2. Solving Factorable Equations
Key idea: When an equation is factored, the solutions come from setting each factor equal to zero.
Steps:
- Move everything to one side.
- Expand if needed.
- Combine like terms.
- Factor.
- Set each factor equal to zero.
Example:
Solve $(x - 3)(x + 5) = -15$
- Move terms: $(x - 3)(x + 5) + 15 = 0$
- Expand: $x^2 + 2x - 15 + 15 = 0 \to x^2 + 2x = 0$
- Factor: $x(x + 2) = 0$
- Solutions: $x = 0$, $x = -2$.
3. Finding x-Intercepts of a Polynomial
Key idea: x-intercepts occur where $y = 0$. Solve by setting each factor equal to zero.
Example:
$y = (5x + 7)(2x - 1)(x - 4)$
- Set $y = 0$
- Solve factors:
- $5x+7 = 0 \to x = -\frac{7}{5}$
- $2x-1 = 0 \to x = \frac{1}{2}$
- $x-4 = 0 \to x = 4$
- x-intercepts: $-\frac{7}{5}, \frac{1}{2}, 4$.
4. Using Graphs to Understand Polynomial Degree and Constant Term
Key idea: The shape of the graph tells you the polynomial's degree behavior.
Even vs. Odd Degree:
- Even-degree polynomials: both ends go in the same direction.
- Odd-degree polynomials: ends go in opposite directions.
Positive vs. Negative Leading Coefficient:
- Ends up = positive leading coefficient.
- Ends down = negative leading coefficient.
Constant Term:
- The constant term is the y-value at $x = 0$ — the y-intercept.
Example:
Graph shows:
- Both ends down → even degree, negative leading coefficient.
- y-intercept at $-4$ → constant term is $-4$.
5. General Strategies to Rely On
- Always look for common factors before expanding.
- When solving, move everything to one side and factor.
- For x-intercepts, set $y = 0$ and solve factors.
- Use graph behavior to determine degree and sign of leading coefficient.
- The y-intercept gives the constant term.