Honors STAG 10 Midterm

Chapter 4: Quadratic Functions and Factoring

Formulas:

  1. Vertex Form:
    y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k

    • Vertex: (h,k)(h, k)(h,k)

    • Axis of Symmetry: x=hx = hx=h

  2. Standard Form:
    y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c

    • Axis of Symmetry: x=−b2ax = -\frac{b}{2a}x=−2ab​

  3. Quadratic Formula:
    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}x=2a−b±b2−4ac​​

  4. Discriminant:
    Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

    • Δ>0\Delta > 0Δ>0: Two real roots

    • Δ=0\Delta = 0Δ=0: One real root

    • Δ<0\Delta < 0Δ<0: No real roots (complex roots)

  5. Factoring:
    For ax2+bx+cax^2 + bx + cax2+bx+c:

    • Look for mmm and nnn such that m⋅n=acm \cdot n = acm⋅n=ac and m+n=bm + n = bm+n=b.

Chapter 5: Polynomials and Polynomial Functions

Formulas:

  1. End Behavior:
    For y=axny = ax^ny=axn:

    • If nnn is even, a>0a > 0a>0, both ends up; a<0a < 0a<0, both ends down.

    • If nnn is odd, a>0a > 0a>0, left down and right up; a<0a < 0a<0, left up and right down.

  2. Remainder Theorem:
    f(a)=Rf(a) = Rf(a)=R, where RRR is the remainder when f(x)f(x)f(x) is divided by x−ax - ax−a.

  3. Factor Theorem:
    x−ax - ax−a is a factor of f(x)f(x)f(x) if f(a)=0f(a) = 0f(a)=0.

Chapter 6: Rational Exponents and Radical Functions

Formulas:

  1. Radical and Exponent Conversion:
    am/n=amna^{m/n} = \sqrt[n]{a^m}am/n=nam​.

  2. Simplify Radicals:

    • Combine like terms under the radical.

    • Rationalize denominators (no radicals in the denominator).

  3. Domain of Radicals:

    • Even roots: Radicand ≥0\geq 0≥0.

    • Odd roots: Radicand ∈R\in \mathbb{R}∈R.

Chapter 7: Exponential and Logarithmic Functions

Formulas:

  1. Exponential Growth and Decay:
    y=abx−h+ky = ab^{x-h} + ky=abx−h+k, where:

    • b>1b > 1b>1: Growth

    • 0<b<10 < b < 10<b<1: Decay

  2. Compound Interest:
    A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}A=P(1+nr​)nt

    • PPP: Principal, rrr: Annual rate, nnn: Number of times compounded per year, ttt: Time in years

  3. Continuous Compound Interest:
    A=PertA = Pe^{rt}A=Pert.

  4. Logarithmic Properties:

    • log⁡(ab)=log⁡a+log⁡b\log(ab) = \log a + \log blog(ab)=loga+logb

    • log⁡(a/b)=log⁡a−log⁡b\log(a/b) = \log a - \log blog(a/b)=loga−logb

    • log⁡(ab)=blog⁡a\log(a^b) = b\log alog(ab)=bloga.

  5. Change of Base Formula:
    log⁡ab=log⁡blog⁡a\log_a b = \frac{\log b}{\log a}loga​b=logalogb​.

Key Terms

  • Parabola: The graph of a quadratic function.

  • Roots/Zeros: Solutions of f(x)=0f(x) = 0f(x)=0.

  • Vertex: The highest or lowest point on a parabola.

  • Axis of Symmetry: A vertical line that divides the parabola into two mirror images.

  • Degree of a Polynomial: The highest power of xxx.

  • Leading Coefficient: The coefficient of the highest degree term.

  • Exponential Function: A function where the variable is in the exponent.

  • Logarithm: The inverse of an exponential function.