AP®︎ Precalculus: Unit 2 Practice Test

Essential Knowledge: 2.1.A.1

A sequence is a function from the whole numbers to the real numbers. Consequently, the graph of a sequence consists of discrete points instead of a curve.

Essential Knowledge: 2.1.A.2

Successive terms in an arithmetic sequence have a common difference, or constant rate of change.

Essential Knowledge: 2.1.A.3

The general term of an arithmetic sequence with a common difference $d$ is denoted by $a_n$ and is given by $a_n = a_0 + dn$, where $a_0$ is the initial value, or by $a_n = a_k + d(n - k)$, where $a_k$ is the $k$th term of the sequence.

Essential Knowledge: 2.1.B.1

Successive terms in an increasing or decreasing geometric sequence have a common ratio, or constant proportional change.

Essential Knowledge: 2.1.B.2

The general term of an increasing or decreasing geometric sequence with a common ratio $r$ is denoted by $g_n$ and is given by $g_n = g_0r^n$, where $g_0$ is the initial value, or by $g_n = g_kr^{(n-k)}$, where $g_k$ is the $k$th term of the sequence.

Essential Knowledge: 2.1.B.3

An increasing arithmetic sequence increases equally with each step, whereas an increasing geometric sequence of positive values increases by a larger amount with each successive step.

Essential Knowledge: 2.2.A.1

Linear functions of the form $f(x) = b + mx$ are similar to arithmetic sequences of the form $a_n = a_0 + dn$, as both can be expressed as an initial value ($b$ or $a_0$) plus repeated addition of a constant rate of change, the slope ($m$ or $d$).

Essential Knowledge: 2.2.A.2

Similar to arithmetic sequences of the form $a_n = a_k + d(n - k)$, which are based on a known difference, $d$, and a $k$th term, linear functions can be expressed in the form $f(x) = y_i + m(x - x_i)$ based on a known slope, $m$, and a point, $(x_i, y_i)$.

Essential Knowledge: 2.2.A.3

Exponential functions of the form $f(x) = ab^x$ are similar to geometric sequences of the form $g_n = g_0r^n$, as both can be expressed as an initial value ($a$ or $g_0$) times repeated multiplication by a constant ratio ($b$ or $r$).

Essential Knowledge: 2.2.A.4

Similar to geometric sequences of the form $g_n = g_kr^{(n-k)}$, which are based on a known ratio, $r$, and a $k$th term, exponential functions can be expressed in the form $f(x) = y_i b^{(x-x_i)}$ based on a known ratio, $b$, and a point, $(x_i, y_i)$.

Essential Knowledge: 2.2.A.5

Sequences and their corresponding functions may have different domains.

Essential Knowledge: 2.2.B.1

Over equal-length input-value intervals, if the output values of a function change at constant rate, then the function is linear; if the output values of a function change proportionally, then the function is exponential. For an exponential function, the known ratio cannot be 0 or 1.

Essential Knowledge: 2.2.B.2

Linear functions of the form $f(x) = b + mx$ and exponential functions of the form $f(x) = ab^x$ can both be expressed analytically in terms of an initial value and a constant involved with change. However, linear functions are based on addition, while exponential functions are based on multiplication.

Essential Knowledge: 2.2.B.3

Arithmetic sequences, linear functions, geometric sequences, and exponential functions all have the property that they can be determined by two distinct sequence or function values.

Essential Knowledge: 2.3.A.1

The general form of an exponential function is $f(x) = ab^x$, with the initial value $a$, where $a \ne 0$, and the base $b$, where $b > 0$, and $b \ne 1$. When $a > 0$ and $b > 1$, the exponential function is said to demonstrate exponential growth. When $a > 0$ and $0 < b < 1$, the exponential function is said to demonstrate exponential decay.

Essential Knowledge: 2.3.A.2

When the natural numbers are input values in an exponential function, the input value specifies the number of factors of the base to be applied to the function’s initial value. The domain of an exponential function is all real numbers.

Essential Knowledge: 2.3.A.3

Because the output values of exponential functions in general form are proportional over equal-length input-value intervals, exponential functions are always increasing or always decreasing, and their graphs are always concave up or always concave down. Consequently, exponential functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.

Essential Knowledge: 2.3.A.4

If the output values of the function $f$ are not proportional over equal-length input-value intervals, but the output values of an additive transformation of $f$ are proportional over equal-length input-value intervals, then $f$ can be modeled by an additive transformation of an exponential function.

Essential Knowledge: 2.3.A.5

For an exponential function in general form, as the input values increase or decrease without bound, the output values will increase or decrease without bound or will get arbitrarily close to zero. That is, for an exponential function in general form, $\lim_{x \to \pm\infty} ab^x = \infty$, $\lim_{x \to \pm\infty} ab^x = -\infty$, or $\lim_{x \to \pm\infty} ab^x = 0$.

Essential Knowledge: 2.4.A.1

The product property for exponents states that $b^m b^n = b^{(m+n)}$. Graphically, this property implies that every horizontal translation of an exponential function, $f(x) = b^{(x+k)}$, is equivalent to a vertical dilation, $f(x) = b^{(x+k)} = b^x b^k = ab^x$, where $a = b^k$.

Essential Knowledge: 2.4.A.2

The power property for exponents states that $(b^m)^n = b^{(mn)}$. Graphically, this property implies that every horizontal dilation of an exponential function, $f(x) = b^{(cx)}$, is equivalent to a change of the base of an exponential function, $f(x) = (b^c)^x$, where $b^c$ is a constant and $c \ne 0$.

Essential Knowledge: 2.4.A.3

The negative exponent property states that $b^{-n} = \frac{1}{b^n}$.

Essential Knowledge: 2.4.A.4

The value of an exponential expression involving an exponential unit fraction, such as $b^{(1/k)}$ where $k$ is a natural number, is the $k$th root of $b$, when it exists.

Essential Knowledge: 2.5.A.1

Exponential functions model growth patterns where successive output values over equal-length input-value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.

Essential Knowledge: 2.5.A.2

A constant may need to be added to the dependent variable values of a data set to reveal a proportional growth pattern.

Essential Knowledge: 2.5.A.3

An exponential function model can be constructed from an appropriate ratio and initial value or from two input-output pairs. The initial value and the base can be found by solving a system of equations resulting from the two input-output pairs.

Essential Knowledge: 2.5.A.4

Exponential function models can be constructed by applying transformations to $f(x) = ab^x$ based on characteristics of a contextual scenario or data set.

Essential Knowledge: 2.5.A.5

Exponential function models can be constructed for a data set with technology using exponential regressions.

Essential Knowledge: 2.5.A.6

The natural base $e$, which is approximately 2.718, is often used as the base in exponential functions that model contextual scenarios.

Essential Knowledge: 2.5.B.1

For an exponential model in general form $f(x) = ab^x$, the base of the exponent, $b$, can be understood as a growth factor in successive unit changes in the input values and is related to a percent change in context.

Essential Knowledge: 2.5.B.2

Equivalent forms of an exponential function can reveal different properties of the function. For example, if $d$ represents number of days, then the base of $f(d) = 2^d$ indicates that the quantity increases by a factor of 2 every day, but the equivalent form $f(d) = (2^7)^{(d/7)}$ indicates that the quantity increases by a factor of $2^7$ every week.

Essential Knowledge: 2.5.B.3

Exponential models can be used to predict values for the dependent variable, depending on the contextual constraints on the domain, and may be used to determine values of the independent variable.

Essential Knowledge: 2.6.A.1

Two variables in a data set that demonstrate a slightly changing rate of change can be modeled by linear, quadratic, and exponential function models.

Essential Knowledge: 2.6.A.2

Models can be compared based on contextual clues and applicability to determine which model is most appropriate.

Essential Knowledge: 2.6.B.1

For a given value of an independent variable, a residual of a regression is the actual value of the dependent variable value minus the value predicted by the function regression model. A residual plot is a graphical representation of the residuals of a regression, with the residuals on the vertical axis and the independent variable on the horizontal axis. A model is justified as appropriate for a data set if the residual plot appears without pattern.

Essential Knowledge: 2.6.B.2

For given values of an independent variable, the errors in a model can relate to the signed differences in the values of the dependent variable predicted by the function regression model and the actual values OR the unsigned absolute values of the residuals. Errors may reveal that the model underestimates or overestimates actual values on a given input-value interval. The data set and context can provide clues as to whether underestimates or overestimates are preferred on a given input-value interval.

Essential Knowledge: 2.7.A.1

If $f$ and $g$ are functions, the composite function $f \circ g$ maps a set of input values to a set of output values such that the output values of $g$ are used as input values of $f$. For this reason, the domain of the composite function is restricted to those input values of $g$ for which the corresponding output value is in the domain of $f$. $(f \circ g)(x)$ can also be represented as $f(g(x))$.

Essential Knowledge: 2.7.A.2

Values for the composite function $f \circ g$ can be calculated or estimated from the graphical, numerical, analytical, or verbal representations of $f$ and $g$ by using output values from $g$ as input values for $f$.

Essential Knowledge: 2.7.A.3

The composition of functions is not commutative; that is, $f \circ g$ and $g \circ f$ are typically different functions; therefore, $f(g(x))$ and $g(f(x))$ are typically different values.

Essential Knowledge: 2.7.A.4

If the function $f(x)=x$ is composed with any function $g$, the resulting composite function is the same as $g$; that is, $g(f(x)) = f(g(x)) = g(x)$. The function $f(x)=x$ is called the identity function. When composing two functions, the identity function acts in the same way as 0, the additive identity, when adding two numbers and 1, the multiplicative identity, when multiplying two numbers.

Essential Knowledge: 2.7.B.1

Function composition is useful for relating two quantities that are not directly related by an existing formula.

Essential Knowledge: 2.7.B.2

When analytic representations of the functions $f$ and $g$ are available, an analytic representation of $f(g(x))$ can be constructed by substituting $g(x)$ for every instance of $x$ in $f$.

Essential Knowledge: 2.7.B.3

A numerical or graphical representation of $f \circ g$ can often be constructed by calculating or estimating values for $(x, f(g(x)))$.

Essential Knowledge: 2.7.C.1

Functions given analytically can often be decomposed into less complicated functions. When properly decomposed, the variable in one function should replace each instance of the function with which it was composed.

Essential Knowledge: 2.7.C.2

An additive transformation of a function, $f$, that results in vertical and horizontal translations can be understood as the composition of $g(x)=x+k$ with $f$.

Essential Knowledge: 2.7.C.3

A multiplicative transformation of a function, $f$, that results in vertical and horizontal dilations can be understood as the composition of $g(x)=kx$ with $f$.

Essential Knowledge: 2.8.A.1

On a specified domain, a function, $f$, has an inverse function, or is invertible, if each output value of $f$ is mapped from a unique input value. The domain of a function may be restricted in many ways to make the function invertible.

Essential Knowledge: 2.8.A.2

An inverse function can be thought of as a reverse mapping of the function. An inverse function, $f^{-1}$, maps the output values of a function, $f$, on its invertible domain to their corresponding input values; that is, if $f(a)=b$, then $f^{-1}(b)=a$. Alternately, on its invertible domain, if a function consists of input-output pairs $(a,b)$, then the inverse function consists of input-output pairs $(b,a)$.

Essential Knowledge: 2.8.B.1

The composition of a function, $f$, and its inverse function, $f^{-1}$, is the identity function; that is, $f(f^{-1}(x)) = f^{-1}(f(x)) = x$.

Essential Knowledge: 2.8.B.2

On a function’s invertible domain, the function’s range and domain are the inverse function’s domain and range, respectively. The inverse of the table of values of $y=f(x)$ can be found by reversing the input-output pairs; that is, $(a,b)$ corresponds to $(b,a)$.

Essential Knowledge: 2.8.B.3

The inverse of the graph of the function $y=f(x)$ can be found by reversing the roles of the $x$- and $y$-axes; that is, by reflecting the graph of the function over the graph of the identity function $h(x)=x$.

Essential Knowledge: 2.8.B.4

The inverse of the function can be found by determining the inverse operations to reverse the mapping. One method for finding the inverse of the function $f$ is reversing the roles of $x$ and $y$ in the equation $y=f(x)$, then solving for $y=f^{-1}(x)$.

Essential Knowledge: 2.8.B.5

In addition to limiting the domain of a function to obtain an inverse function, contextual restrictions may also limit the applicability of an inverse function.

Essential Knowledge: 2.9.A.1

The logarithmic expression $\log_b c$ is equal to, or represents, the value that the base $b$ must be exponentially raised to in order to obtain the value $c$. That is, $\log_b c = a$ if and only if $b^a = c$, where $a$ and $c$ are constants, $b > 0$, and $b \ne 1$. (when the base of a logarithmic expression is not specified, it is understood as the common logarithm with a base of 10)

Essential Knowledge: 2.9.A.2

The values of some logarithmic expressions are readily accessible through basic arithmetic while other values can be estimated through the use of technology.

Essential Knowledge: 2.9.A.3

On a logarithmic scale, each unit represents a multiplicative change of the base of the logarithm. For example, on a standard scale, the units might be 0, 1, 2, ..., while on a logarithmic scale, using logarithm base 10, the units might be $10^0$, $10^1$, $10^2$, ....

Essential Knowledge: 2.10.A.1

The general form of a logarithmic function is $f(x) = a\log_b x$, with base $b$, where $b > 0$, $b \ne 1$, and $a \ne 0$.

Essential Knowledge: 2.10.A.2

The way in which input and output values vary together have an inverse relationship in exponential and logarithmic functions. Output values of general-form exponential functions change proportionately as input values increase in equal-length intervals. However, input values of general-form logarithmic functions change proportionately as output values increase in equal-length intervals. Alternately, exponential growth is characterized by output values changing multiplicatively as input values change additively, whereas logarithmic growth is characterized by output values changing additively as input values change multiplicatively.

Essential Knowledge: 2.10.A.3

$f(x) = \log_b x$ and $g(x) = b^x$, where $b > 0$ and $b \ne 1$, are inverse functions. That is, $g(f(x)) = f(g(x)) = x$.

Essential Knowledge: 2.10.A.4

The graph of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \ne 1$, is a reflection of the graph of the exponential function $g(x) = b^x$, where $b > 0$ and $b \ne 1$, over the graph of the identity function $h(x) = x$.

Essential Knowledge: 2.10.A.5

If $(s, t)$ is an ordered pair of the exponential function $g(x) = b^x$, where $b > 0$ and $b \ne 1$, then $(t, s)$ is an ordered pair of the logarithmic function $f(x) = \log_b x$, where $b > 0$ and $b \ne 1$.

Essential Knowledge: 2.11.A.1

The domain of a logarithmic function in general form is any real number greater than zero, and its range is all real numbers.

Essential Knowledge: 2.11.A.2

Because logarithmic functions are inverses of exponential functions, logarithmic functions are also always increasing or always decreasing, and their graphs are either always concave up or always concave down. Consequently, logarithmic functions do not have extrema except on a closed interval, and their graphs do not have points of inflection.

Essential Knowledge: 2.11.A.3

The additive transformation function $g(x)=f(x+k)$, where $k\ne 0$, of a logarithmic function $f$ in general form does not have input values that are proportional over equal-length output-value intervals. However, if the input values of the additive transformation function, $g(x)=f(x+k)$, of any function $f$ are proportional over equal-length output value intervals, then $f$ is logarithmic.

Essential Knowledge: 2.11.A.4

With their limited domain, logarithmic functions in general form are vertically asymptotic to $x=0$, with an end behavior that is unbounded. That is, for a logarithmic function in general form, $\lim_{x\to 0^+}a\log_b x=\pm\infty$ and $\lim_{x\to\infty}a\log_b x=\pm\infty$.

Essential Knowledge: 2.12.A.1

The product property for logarithms states that $\log_{b}(xy) = \log_{b} x + \log_{b} y$. Graphically, this property implies that every horizontal dilation of a logarithmic function, $f(x) = \log_{b}(kx)$, is equivalent to a vertical translation, $f(x) = \log_{b}(kx) = \log_{b} k + \log_{b} x = a + \log_{b} x$, where $a = \log_{b} k$.

Essential Knowledge: 2.12.A.2

The power property for logarithms states that $\log_{b} x^{n} = n\log_{b} x$. Graphically, this property implies that raising the input of a logarithmic function to a power, $f(x) = \log_{b} x^{k}$, results in a vertical dilation, $f(x) = \log_{b} x^{k} = k\log_{b} x$.

Essential Knowledge: 2.12.A.3

The change of base property for logarithms states that $\log_{b} x = \frac{\log_{a} x}{\log_{a} b}$, where $a > 0$ and $a \ne 1$. This implies that all logarithmic functions are vertical dilations of each other.

Essential Knowledge: 2.12.A.4

The function $f(x) = \ln x$ is a logarithmic function with the natural base $e$; that is, $\ln x = \log_{e} x$.

Essential Knowledge: 2.13.A.1

Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can be used to solve equations and inequalities involving exponents and logarithms.

Essential Knowledge: 2.13.A.2

When solving exponential and logarithmic equations found through analytical or graphical methods, the results should be examined for extraneous solutions precluded by the mathematical or contextual limitations.

Essential Knowledge: 2.13.A.3

Logarithms can be used to rewrite expressions involving exponential functions in different ways that may reveal helpful information. Specifically, $b^x = c^{(\log_c b)(x)}$.

Essential Knowledge: 2.13.B.1

The function $f(x)=ab^{(x+h)}+k$ is a combination of additive transformations of an exponential function in general form. The inverse of $y=f(x)$ can be found by determining the inverse operations to reverse the mapping.

Essential Knowledge: 2.13.B.2

The function $f(x)=a\log_b(x+h)+k$ is a combination of additive transformations of a logarithmic function in general form. The inverse of $y=f(x)$ can be found by determining the inverse operations to reverse the mapping.

Essential Knowledge: 2.14.A.1

Logarithmic functions are inverses of exponential functions and can be used to model situations involving proportional growth, or repeated multiplication, where the input values change proportionally over equal-length output-value intervals. Alternately, if the output value is a whole number, it indicates how many times the initial value has been multiplied by the proportion.

Essential Knowledge: 2.14.A.2

A logarithmic function model can be constructed from an appropriate proportion and a real zero or from two input-output pairs.

Essential Knowledge: 2.14.A.3

Logarithmic function models can be constructed by applying transformations to $f(x) = a\log_b x$ based on characteristics of a context or data set.

Essential Knowledge: 2.14.A.4

Logarithmic function models can be constructed for a data set with technology using logarithmic regressions.

Essential Knowledge: 2.14.A.5

The natural logarithm function is often useful in modeling real-world phenomena.

Essential Knowledge: 2.14.A.6

Logarithmic function models can be used to predict values for the dependent variable, and they may be used to determine values of the independent variable.

Essential Knowledge: 2.15.A.1

In a semi-log plot, one of the axes is logarithmically scaled with logarithm base $n$, where $n > 1$. When the $y$-axis of a semi-log plot is logarithmically scaled, data or functions that demonstrate exponential characteristics will appear linear.

Essential Knowledge: 2.15.A.2

If for large input values of a bivariate data set the logarithms of the output values trend linear, then transformations of an exponential function can be used to model the data set. Therefore, an advantage of semi-log plots is that the dependent variable values do not need to be transformed by a constant to reveal that transformations of an exponential function would produce an appropriate model.

Essential Knowledge: 2.15.B.1

Techniques used to model linear functions can be applied to a semi-log graph.

Essential Knowledge: 2.15.B.2

For an exponential model of the form $y = ab^x$, the corresponding linear model for the semi-log plot is $y = (\log_n b)x + \log_n a$, where $n > 0$ and $n \ne 1$. Specifically, the linear rate of change is $\log_n b$, and the initial linear value is $\log_n a$.