Part 2 of the 6-part TExES Mathematics 7-12 (235) refresher series (see Part 1: Number Concepts). This is the largest domain on the exam, spanning functions, algebra, exponentials and logarithms, trigonometry, and an introduction to calculus (Competencies 004-010).
- Work with function basics: domain and range, even/odd symmetry, transformations, inverses, and composition.
- Analyze linear and quadratic functions, including the quadratic formula and the discriminant.
- Apply the Fundamental Theorem of Algebra and the Rational Root Theorem to polynomial, rational, and radical functions.
- Convert between exponential and logarithmic form and apply the log rules.
- Use the unit circle, trig identities, and the Law of Sines and Law of Cosines.
- Apply basic differentiation and integration rules, including the Fundamental Theorem of Calculus.
Functions: The Basics
A relation is a function exactly when every input maps to exactly one output, which is what the vertical line test checks on a graph. The domain is the set of valid inputs, and the range is the set of valid outputs.
A function is even when \( f(-x) = f(x) \), giving symmetry about the \( y \)-axis, and odd when \( f(-x) = -f(x) \), giving symmetry about the origin. The standard transformations of \( f(x) \): adding \( k \) shifts the graph vertically, replacing \( x \) with \( x-h \) shifts it horizontally (the sign flips the direction you'd expect), multiplying by \( a \) stretches or compresses it vertically, \( f(-x) \) reflects over the \( y \)-axis, and \( -f(x) \) reflects over the \( x \)-axis.
To find an inverse function, swap \( x \) and \( y \) and solve for \( y \); \( f \) and \( f^{-1} \) are always reflections of each other over the line \( y = x \). Composition \( (f \circ g)(x) = f(g(x)) \) means apply \( g \) first, then \( f \).
Linear and Quadratic Functions
Linear functions come in three common forms: slope-intercept \( y = mx+b \), point-slope \( y - y_1 = m(x-x_1) \), and standard \( Ax+By=C \). Slope, \( m = \frac{y_2-y_1}{x_2-x_1} \), is the rate of change; parallel lines share a slope, and perpendicular lines have slopes that are negative reciprocals.
For quadratics, standard form is \( f(x) = ax^2+bx+c \) and vertex form is \( f(x) = a(x-h)^2+k \), with vertex \( (h,k) \) and \( h = -\frac{b}{2a}\). The quadratic formula solves for the roots directly:
\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]The discriminant \( b^2-4ac \) tells you what kind of roots to expect before you even finish solving: positive gives two real roots, zero gives one repeated real root, and negative gives two complex conjugate roots. The parabola opens upward, with a minimum, when \( a>0 \), and downward, with a maximum, when \( a<0 \).
Polynomial, Rational, Radical & Piecewise Functions
A degree-\( n \) polynomial has exactly \( n \) roots total, counting complex roots and multiplicity, per the Fundamental Theorem of Algebra. End behavior is controlled entirely by the leading term: its degree (even or odd) and the sign of its coefficient. The Rational Root Theorem narrows the search for rational roots to
\[ \pm \frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}} \]For rational functions, vertical asymptotes occur wherever the denominator is 0 and the numerator isn't also 0 there. Horizontal asymptotes come from comparing degrees: numerator degree less than denominator gives \( y=0 \), equal degrees give \( y \) equal to the ratio of leading coefficients, and numerator degree greater gives no horizontal asymptote (possibly a slant one instead). Radical functions with an even index restrict the domain so the radicand is nonnegative. Absolute value functions are V-shaped, with \( |x-h|+k \) vertexed at \( (h,k) \). Piecewise functions use different expressions on different intervals, so always check whether the pieces connect at the breakpoints.
Exponential and Logarithmic Functions
An exponential model \( f(x) = a \cdot b^x \) grows when \( b>1 \) and decays when \( 0<b<1 \), with a horizontal asymptote at \( y=0 \) unless it's been shifted. A logarithm is simply the inverse of an exponential: \( y = \log_b(x) \) says exactly the same thing as \( b^y = x \). The log rules follow from that relationship:
\[ \log_b(MN) = \log_b M + \log_b N, \qquad \log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N, \qquad \log_b(M^p) = p\log_b M \]and changing base uses \( \log_b M = \frac{\ln M}{\ln b} \), where \( e \approx 2.71828 \) and \( \ln(x) = \log_e(x) \). A common application is compound interest, \( A = P\left(1+\frac{r}{n}\right)^{nt} \), with continuous growth as the limiting case, \( A = Pe^{rt} \).
Trigonometric and Circular Functions
On the unit circle, a point at angle \( \theta \) has coordinates \( (\cos\theta, \sin\theta) \), and \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). The Pythagorean identity, \( \sin^2\theta+\cos^2\theta=1 \), is the one to know cold, along with the reciprocal identities \( \csc\theta = \frac{1}{\sin\theta} \), \( \sec\theta = \frac{1}{\cos\theta} \), \( \cot\theta = \frac{1}{\tan\theta} \). Worth memorizing outright: \( \sin 30^\circ = \frac12 \), \( \sin 45^\circ = \frac{\sqrt2}{2} \), \( \sin 60^\circ = \frac{\sqrt3}{2} \), with cosine mirroring these in reverse.
Both \( y=\sin x \) and \( y=\cos x \) have period \( 2\pi \) and amplitude 1, while \( y=\tan x \) has period \( \pi \) and vertical asymptotes. For a general sinusoid \( y = A\sin(Bx-C)+D \), the amplitude is \( |A| \), the period is \( \frac{2\pi}{|B|} \), the phase shift is \( \frac{C}{B} \), and \( D \) is the vertical shift. For non-right triangles, reach for the Law of Sines, \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), or the Law of Cosines, \( c^2 = a^2+b^2-2ab\cos C \).
Differential and Integral Calculus
The derivative is the instantaneous rate of change, the slope of the tangent line at a point:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]The power rule makes most derivatives quick: \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \). The sign of \( f' \) tells you where a function is increasing or decreasing, and the sign of \( f'' \) tells you concavity; critical points, where \( f'(x)=0 \) or is undefined, locate local maxima and minima.
The integral represents accumulation, the area under a curve, and the Fundamental Theorem of Calculus ties the two ideas together:
\[ \int_a^b f'(x)\,dx = f(b) - f(a) \]The basic power rule for integration runs in reverse: \( \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \), for \( n \neq -1 \).
Exercises
- The function \( f(x) = -2(x-3)^2+5 \) has its vertex at: A) \( (3,5) \) B) \( (-3,5) \) C) \( (3,-5) \) D) \( (-2,3) \)
- What is the horizontal asymptote of \( f(x) = \frac{3x^2+1}{x^2-4} \)? A) \( y=0 \) B) \( y=3 \) C) \( y=-4 \) D) No horizontal asymptote
- If \( \log_2(x) = 5 \), what is \( x \)? A) 10 B) 25 C) 32 D) 2.5
- Which value of \( \theta \) (\( 0 \le \theta < 2\pi \)) satisfies \( \sin\theta = -\frac12 \) in the third quadrant? A) \( \pi/6 \) B) \( 7\pi/6 \) C) \( 5\pi/6 \) D) \( 11\pi/6 \)
- Find \( f'(x) \) for \( f(x) = 4x^3-2x+7 \). A) \( 12x^2-2 \) B) \( 4x^2-2x \) C) \( 12x^3-2x \) D) \( 12x^2+7 \)
- A population grows according to \( P(t) = 500e^{0.03t} \). What does 0.03 represent? A) The initial population B) The continuous growth rate (3% per unit of time) C) The time elapsed D) The doubling time
- 1. A. Vertex form \( a(x-h)^2+k \) has vertex \( (h,k) \), so the vertex is \( (3,5) \).
- 2. B. Numerator and denominator have equal degree (both 2), so the asymptote is the ratio of leading coefficients: \( 3/1=3 \).
- 3. C. \( \log_2(x)=5 \) means \( 2^5=x \), and \( 2^5=32 \).
- 4. B. The reference angle is \( \pi/6 \). In the third quadrant that becomes \( \pi+\pi/6 = 7\pi/6 \).
- 5. A. \( \frac{d}{dx}(4x^3)=12x^2 \), \( \frac{d}{dx}(-2x)=-2 \), \( \frac{d}{dx}(7)=0 \).
- 6. B. In continuous growth models \( Pe^{rt} \), \( r \) is the continuous growth rate.
Continue to Part 3: Geometry and Measurement →, covering the axiomatic structure of Euclidean geometry, triangle and circle theorems, and coordinate and vector geometry.
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