Study guide: Quadratic Functions and Factoring

📋 Course Outline

1. Forms of Quadratic Functions
2. Expanding and Changing Forms
3. Common Factors and Trinomials
4. Special Factoring Patterns
5. Solving Quadratics by Factoring
6. Quadratic Graphs and Intervals

📖 1. Forms of Quadratic Functions

★ Must-know

• In standard form, $y=ax^2+bx+c$, the leading coefficient a determines the parabola’s opening direction and shape, while c gives its y-intercept.

• In vertex form, $y=a(x-h)^2+k$, the vertex is (h, k), and a gives information about opening direction and shape.

• In factored form, the values that make the factors zero are the zeros or x-intercepts of the quadratic.

Further detail

📐 Formula - The axis of symmetry of $y=ax^2+bx+c$ is $x=-\frac{b}{2a}$.

💡 Memory Hook

Standard form reveals the y-intercept; vertex form displays the vertex.

📖 2. Expanding and Changing Forms

★ Must-know

• 🔄 To expand a squared binomial:
  1. Write the bracket twice as a product of identical factors.
  2. Multiply using FOIL.
  3. Simplify using the order of operations.

Further detail

• 🔄 To convert factored form to standard form:
  1. Expand the brackets using FOIL.
  2. Distribute any coefficient outside the brackets.
  3. Collect like terms.

💡 Memory Hook

FOIL: First, Outside, Inside, Last.

📖 3. Common Factors and Trinomials

📝 Essential Points

• The greatest common factor of polynomial terms uses the greatest common coefficient and the lowest exponent of each variable shared by the terms.

• For a trinomial with leading coefficient 1, factor c into pairs and choose the pair whose sum or difference gives the middle coefficient b; use the pair as the constants in the two binomials.

• For a trinomial $ax^2+bx+c$ with $a\ne1$, multiply a and c, split the middle term into terms whose coefficients multiply to ac and add to b, group the four terms, factor each group to obtain a common binomial, and factor that binomial once.

💡 Memory Hook

For ax² + bx + c, split the middle term, group, then factor the shared binomial.

📖 4. Special Factoring Patterns

★ Must-know

📌 A difference of two squares is a binomial with a subtraction in which both terms are perfect squares, and it factors as the difference and sum of their square roots.

📐 Formula - A difference of squares factors according to $A^2-B^2=(A+B)(A-B)$.

📌 A perfect-square trinomial has positive perfect-square first and last terms, and its middle term equals twice the product of their square roots; it factors as the square of a binomial with the sign of the middle term.

Further detail

• The trinomial $9x^2-12x+4$ is a perfect-square trinomial and factors as $(3x-2)^2$.

💡 Memory Hook

Difference of squares uses opposite signs; a perfect-square trinomial repeats the same binomial.

📖 5. Solving Quadratics by Factoring

★ Must-know

• 🔄 Solve a quadratic equation by factoring:
  1. Put all terms on one side and set the equation equal to zero.
  2. Factor the quadratic.
  3. Set each factor equal to zero.
  4. Solve each resulting equation.

Further detail

• For $x^2-5x-50=0$, factoring gives $(x+10)(x-5)=0$, so the roots are $x=-10$ and $x=5$.

• For $h(t)=-5t^2+10t+70$, factoring the equation $h(t)=0$ gives $-5(t-4)(t+2)=0$, with zeros $t=4$ and $t=-2$.

💡 Memory Hook

Set equal to zero, factor, set each factor to zero, solve.

📖 6. Quadratic Graphs and Intervals

🔑 Key Concepts & Definitions

• Positive and Negative Intervals: Intervals identify the x-values where the quadratic’s y-values are above and below the x-axis, respectively; the interval inequalities are strict.
• Increasing and Decreasing Intervals: An increasing interval is a portion where increasing x-values produce increasing y-values, while a decreasing interval is a portion where increasing x-values produce decreasing y-values.

★ Must-know

• A parabola opens upward when its leading coefficient is positive and downward when its leading coefficient is negative.

• For a quadratic function, the vertex is the point where the graph changes between increasing and decreasing.

Further detail

• The second differences are positive for an upward-opening parabola and negative for a downward-opening parabola.

💡 Memory Hook

Above the x-axis is positive; below it is negative.