Advanced Math: Expressions & Functions

Make sense of the idea

For positive a, $\displaystyle a^{\frac{m}{n}}$ means the nth root of a, raised to power m. The denominator names the root; the numerator names the power.

Choose a method

For $\displaystyle a > 0$, $\displaystyle a^{-r} = \frac{1}{a^{r}}$. When multiplying powers with the same base, add exponents. Do not add the bases.

Check and interpret

Over real numbers, even roots require nonnegative radicands. The principal square root is nonnegative: $\displaystyle \sqrt{x^{2}} = \left|x\right|$, not always x. Keep denominator restrictions when simplifying rational expressions.

Common pitfalls

Explain it to yourself

In rational exponents & radicals, what information determines your method, and how can you check whether your result fits the original conditions?

Make sense of the idea

Write $\displaystyle ax^{2} + bx + c = 0$ with $\displaystyle a \ne 0$. The quadratic formula is $\displaystyle x = \frac{-b \pm \sqrt{b^{2} - 4\, ac}}{2\, a}$. The denominator 2a divides the entire numerator.

Choose a method

To complete the square in $\displaystyle x^{2} + bx$, add $\displaystyle \left(\frac{b}{2}\right)^{2}$. Add it to both sides. For example, $\displaystyle x^{2} + 6\, x = 7$ becomes $\displaystyle \left(x + 3\right)^{2} = 16$, so $\displaystyle x = 1$ or −7.

Check and interpret

The discriminant $\displaystyle D = b^{2} - 4\, ac$ counts real roots: $\displaystyle D > 0$ gives two, $\displaystyle D = 0$ gives one repeated root, $\displaystyle D < 0$ gives none. A context can rule out a negative length without deleting it from the algebraic equation. Parent equation: $\displaystyle y = x^{2}$.

Common pitfalls

Explain it to yourself

In quadratic formula & completing the square, what information determines your method, and how can you check whether your result fits the original conditions?

Make sense of the idea

First list every value that makes an original denominator zero. These values are forbidden even if a factor later cancels.

Choose a method

Multiply both sides by a common denominator only on the allowed domain. Solve the resulting equation, then substitute every candidate into the original equation.

Check and interpret

An equation can reduce to a contradiction, an identity with excluded points, or a finite set of valid roots. A canceled factor does not erase an original hole in the domain.

Common pitfalls

Explain it to yourself

In rational equations & domain restrictions, what information determines your method, and how can you check whether your result fits the original conditions?

Make sense of the idea

The principal square root $\displaystyle \sqrt{u}$ is nonnegative and is real only if $\displaystyle u \ge 0$. In $\displaystyle \sqrt{u} = v$, v must also be nonnegative.

Choose a method

Isolate the radical before squaring. Squaring can lose sign information, so it can introduce candidates that never satisfied the original equation.

Check and interpret

Substitute every candidate back into the unsquared equation. If the left side is a nonnegative root while the right side is negative, reject that candidate.

Common pitfalls

Explain it to yourself

In radical equations & extraneous roots, what information determines your method, and how can you check whether your result fits the original conditions?

Make sense of the idea

Parent equations: $\displaystyle y = x$ for lines and $\displaystyle y = x^{2}$ for parabolas. An intersection must have the same x and y in both equations.

Choose a method

If both equations give y, set their right sides equal. Rearrange into a quadratic and solve. Substitute each x into either original equation to recover y.

Check and interpret

The quadratic discriminant tells you whether the line crosses the parabola twice, touches once, or misses it. The number of intersections is not automatically two.

Common pitfalls

Explain it to yourself

In lines & parabolas: nonlinear systems, what information determines your method, and how can you check whether your result fits the original conditions?

Make sense of the idea

In $\displaystyle A\, \left(t\right) = A_{0}\, b^{t}$, $\displaystyle A_{0}$ is the initial value and b is the factor for one unit of t. Growth of r as a decimal uses $\displaystyle b = 1 + r$; decay uses $\displaystyle b = 1 - r$.

Choose a method

Equal time steps have equal ratios in an exponential model; a linear model has equal differences. Check that the time steps are equal before comparing.

Check and interpret

If a process doubles every 3 hours and t is hours, use $\displaystyle A_{0} \cdot 2^{\frac{t}{3}}$. The denominator converts elapsed hours into the number of doubling periods. Parent family: $\displaystyle y = b^{x}$, $\displaystyle b > 0$ and $\displaystyle b \ne 1$.

Common pitfalls

Explain it to yourself

In exponential growth, decay & tables, what information determines your method, and how can you check whether your result fits the original conditions?