10 | eureka Math alGeBra I Study GuIde
As new concepts are introduced in high school, the overarching theme of functions
remains the same. For example, students study the properties of logarithms (e.g.,
logab=+loglabog ) by graphing functions such as fx()=+logl 3 ogx and gx()=log 3 x,
noticing the graphs are the same, and then proving conjectures about the properties of
logarithms—just as they did when they were discovering relationships between sine and
cosine functions earlier. Thus they build on their knowledge in new but analogous ways. The
following paragraphs describe a few examples of major topics (in addition to those mentioned
previously) in high school mathematics and how functions tie them together:
Sequences
A sequence is usually thought of as an ordered list of numbers, but it really is a function
whose domain is contained within the set of nonnegative integers. Because the domain of a
sequence is discrete, sequences are “simplified functions” that are very useful precursors to
studying the “full version” functions. For example, the exponential function fx()= 2 x generates
the sequence {1, 2, 4, 8, 16,.. .}, which carefully avoids having to discuss what the values of 2
21
or 2π are before students are ready. We use sequences in Algebra I to show how linear and
exponential functions are like each other. (Linear functions have a constant rate of change,
and exponential functions have a constant percent rate of change.) More important, we use
sequences to show how linear and exponential functions differ.
Solving Equations
It is very common in modeling applications to compare values of two functions, f and g, and
ask, “For what number(s) x is fx()=gx() true?” For example, if fx()=-x^23 and gx()= 5 x, the
question reduces to solving the equation xx^2 -= 35. Students were introduced to solving
linear and simple quadratic equations in middle school, but it is modeling with exponential,
trigonometric, and logarithmic functions in high school that provides the rich context in which
students generate new types of equations and learn the value of solving those equations.
Geometry
Geometry in high school is the study of geometric transformations—that is, functions that
map points and figures in the plane to new points and figures in the plane. The most
important geometric transformations are translations, reflections, rotations, and dilations.
Translations, reflections, and rotations are functions that preserve distances, leading to the
notion of congruence and theorems about congruent triangles. Dilations are functions that
preserve the “shape” of figures—that is, dilations map figures to similar figures. It is through
the study of geometric transformations that students generalize triangle congruence and
similarity to any figure or graph in the plane (not just triangles).
transformations of functions
In Algebra I, students use geometric transformations to transform graphs of functions in
exactly the same way they did for triangles. For example, by studying the graph of yf= ()x and
the graph of yf=-()x 3 , students see that the graph of yf=-()x 3 is a translation of the graph
of yf= ()x by 3 units to the right in the Cartesian plane. They continue to use translations and
reflections to study properties of the different types of functions throughout high school. In
this way, geometry and algebra inform each other through the use of functions.
calculus
Calculus is the study of differentiation and integration of functions. It is absolutely essential
that students are comfortable with the main types of functions before starting a calculus