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What is the standard form of a quadratic equation?
$$ax^2 + bx + c = 0$$, where $$a ≠ 0$$
What is the quadratic formula?
$$x = frac{-b ± sqrt{b^2 - 4ac}}{2a}$$
What is the discriminant and what does it indicate?
Discriminant: $$b^2 - 4ac$$. Positive: Two real solutions. Zero: One real solution. Negative: No real solutions.
What is the general form of an exponential function?
$$f(x) = a * b^x$$ where *a* is the initial value and *b* is the growth/decay factor.
What is the product rule for exponents?
$$a^m * a^n = a^{m+n}$$
What is the quotient rule for exponents?
$$a^m / a^n = a^{m-n}$$
What is the power rule for exponents?
$$(a^m)^n = a^{mn}$$
What is the zero exponent rule?
$$a^0 = 1$$
What is the negative exponent rule?
$$a^{-n} = 1 / a^n$$
State the change of base formula for logarithms.
$$log_a(x) = frac{log_b(x)}{log_b(a)}$$
How do you solve a quadratic equation by factoring?
Rewrite the quadratic as a product of two linear factors and set each factor to zero.
How do you determine if an exponential function represents growth or decay?
If the base *b* > 1, it's growth. If 0 < *b* < 1, it's decay.
Explain how to use the quadratic formula.
Identify a, b, and c in $$ax^2 + bx + c = 0$$, then substitute into $$x = frac{-b ± sqrt{b^2 - 4ac}}{2a}$$.
How do you solve for time in a compound interest problem?
Use the formula $$A(t) = P * (1 + r)^t$$ and solve for *t* using logarithms.
How do you interpret the solutions of a quadratic equation in a real-world problem?
Check if the solutions make sense in the context (e.g., no negative lengths). Discard extraneous solutions.
Describe how to set up a quadratic equation for an area problem.
Assign a variable to the unknown dimension, express the other dimension in terms of that variable, and use the area formula to create the equation.
What is the first step in solving an exponential equation with the variable in the exponent?
Take the logarithm of both sides of the equation.
How do you find the initial population in an exponential growth model?
The initial population is the value of the function when t=0, represented by 'a' in $$f(x) = a * b^x$$
How to identify the growth/decay rate from a given exponential function?
In $$f(x) = a * b^x$$, if b = 1 + r, then r is the growth rate. If b = 1 - r, then r is the decay rate.
How do you find the half-life in a radioactive decay problem?
Use the formula $$A(t) = A_0 * (0.5)^{t/h}$$, where h is the half-life. Solve for h when A(t) is half of A_0.