MA001: College Algebra

5a. identify the properties of power, quadratic, and polynomial functions given a graph or an equation including end behavior, domain, and range, and local behaviors by using algebraic operations, interval notation, and words

• Compare the forms of quadratic functions, power functions, and polynomial functions.
• For a quadratic function, identify the shape of the graph, the vertex, and axis of symmetry. Compare the $y$-intercept with the vertex. How does the leading coefficient determine whether the graph opens up or down?
• Compare the domain and range for quadratic functions, power functions, and polynomial functions. 
• For a power function, explain how the exponent on the variable determines the end behavior.
• For a polynomial function, identify the degree and leading coefficient of the function. Identify the end behavior of a polynomial function and the number of turning points. Explain how to connect the turning points to local minimum or local maximum points.

A quadratic function is a function of the form $f(x)=a x^{2}+b x+c$, with $a, b$, and $c$ real numbers and $a \neq 0$; this form of the function is known as the general form of a quadratic function. The graph of a quadratic function is a parabola that opens up if $a > 0$ and opens down if $a < 0$. A quadratic equation can also be represented in the form $f(x)=a(x-h)^{2}+k$, known as the standard form or vertex form of a quadratic function. The ordered pair $(h, k)$ is the vertex of the parabola, that is, the minimum point or the maximum point of the parabola, depending on whether the parabola opens up or down, respectively; the value of $h$ can be found as $h=-\frac{b}{2 a}$. The quadratic function or parabola is symmetric about the vertical line drawn through the vertex; this line is called the axis of symmetry.

The $y$-intercept is the value of $y$ where the parabola crosses the $y$-axis, that is, it is the value of $y$ when $x=0$. Similarly, the $x$-intercept is the value of $x$ where the parabola crosses the $x$-axis, that is, the value of $x$ when $y=0$; the values of the $x$-intercept are also known as the zeros of the quadratic function. The zeros of the quadratic function can be found using the quadratic formula: $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} ;$ there are two zeros if the parabola crosses the $x$-axis in two points, one zero if the vertex is on the $x$-axis, or no zeros if the parabola does not intersect the $x$-axis.

The graph below shows these important points marked on the quadratic function.

The table below compares the domain and range as well as the behavior of the quadratic function depending on whether the parabola opens up or down. 

A power function is a function of the form $f(x)=k x^{p}$ where $k$ and $p$ are real numbers; $k$ is the coefficient of the power function. Notice that in a power function the variable is raised to a power. Both the basic linear function $f(x)=x$ and the basic quadratic function $f(x)=x^{2}$ are examples of power functions.

Power functions in which the exponent is a non-negative integer have specific properties of interest. If the exponent $p$ is even, then the power function is an even function, meaning it is symmetric about the $y$-axis. The function opens up if $k > 0$ and down if $k < 0$. So, the domain is $(-\infty, \infty)$ and the range is $[0, \infty)$. The end behavior of the function is the behavior of the function as $x \rightarrow \infty$ or as $x \rightarrow-\infty$. For an even power function, if $k > 0$ the function approaches $\infty$ as $x \rightarrow \infty$ or as $x \rightarrow-\infty$; in contrast, if $k < 0$, the function approaches $-\infty$ as $x \rightarrow \infty$ or as $x \rightarrow-\infty$.

A similar analysis can be applied to power functions in which the non-negative integer is odd. In this case, the function is an odd function, meaning it is symmetric about the origin. If $k > 0$, then the function approaches $\infty$ as $x \rightarrow \infty$ and approaches $-\infty$ as $x \rightarrow-\infty$; in contrast, if $k < 0$, the function approaches $\infty$ as $x \rightarrow-\infty$ and approaches $-\infty$ as $x \rightarrow \infty$. In this case, both the domain and range are $(-\infty, \, \infty)$.

The graphs below represent an even power function and an odd power function, both for $k > 0$.

A polynomial function is a function of the form $f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{2} x^{2}+a_{1} x+a_{0}$ where $n$ is a non-negative integer, each $a_{i}$ is a real number, and $a_{n} \neq 0$. Every $a_{i} x^{i}$ is called a term of the polynomial. The greatest value of the exponent is the degree of the polynomial and the term with the greatest value of the exponent is the leading term. The end behavior of the polynomial function is determined by the end behavior of the leading term. Notice that the leading term is an even or odd power function, so the end behavior of the polynomial function is the same as the end behavior of an even or odd power function as described above. The domain of a polynomial function is $(-\infty, \infty)$; the range can vary depending on the degree of the polynomial and whether it has any absolute maximum or absolute minimum points.

Polynomial functions are continuous and smooth, meaning there are no breaks in the graph and no sharp corners, respectively. Depending on the degree of the polynomial and the values of the non-leading terms, the polynomial may have various turning points, which are the local minimum and local maximum points, that is, the points at which the polynomial changes from decreasing to increasing or increasing to decreasing, respectively. For a polynomial with degree $n$, there are at most $n-1$ of these turning points. As with other functions, the $y$-intercept is where the polynomial crosses the $y$-axis; the $x$-intercepts or zeros are where the polynomial crosses the $x$-axis. A polynomial with degree $n$ has at most $n$ distinct zeros or $x$-intercepts.

For instance, the polynomial function $f(x)=4 x^{7}-3 x^{5}+4 x^{2}-8$ is a polynomial of degree 7. It has at most 7 zeros and 6 turning points. However, these values are the maximum number of zeros and turning points. As shown by the graph of $y=f(x)$ below, the polynomial function may not have this many zeros or turning points. In this case, the function has 2 turning points, one a local minimum and one a local maximum, and only 1 zero or $x$-intercept. One of the turning points occurs at the point $(0, \, -8)$ associated with the $y$-intercept.

Review material in Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions, Power Functions, and Polynomial Functions.

5b. relate elements of the standard form of a quadratic function with corresponding transformations to its graph

• Describe how the standard or vertex form of a quadratic function provides insights into the use of transformations to graph a quadratic function.

Recall that the standard or vertex form of a quadratic function has the form $f(x)=a(x-h)^{2}+k$. The coefficient $a$ describes the vertical stretch or compression of the graph of $f(x)=x^{2}$, depending on whether $a > 1$ or $0 < a < 1$, respectively. If $a < 0$, the graph is reflected vertically across the $x$-axis, meaning the parabola opens down. The vertex has coordinates $(h, \, k)$ indicating a shift of the function $f(x)=x^{2}$ by $h$ units horizontally and $k$ units vertically.

Consider the function $f(x)=-3(x+1)^{2}+5$. The graph of this function is the graph of $f(x)=x^{2}$ when stretched vertically by a factor of $3$ ; the graph is then reflected vertically across the $x$-axis. Then, because the vertex is at $(-1, \, 5)$, the graph will be shifted 1 unit to the left and 5 units up. For instance, the point $(1, \,  1)$ on $f(x)=x^{2}$ (the red graph) corresponds to $(1,3)$ after the vertical stretch, then corresponds to $(1,-3)$ after reflection across the $x$-axis, and then corresponds to $(0, \, 2)$ after a shift of 1 unit to the left and 5 units up. You can verify that $(0, \, 2)$ is on the final (black) graph of $f(x)=-3(x+1)^{2}+5$.

Review material in Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions.

5c. construct the equation of a power, quadratic, or polynomial function given its graph

• Explain how the characteristics of the graph of a quadratic function can be used to construct an equation for the function.
• Use the properties of the graphs of power functions to construct equations for those power functions.
• Describe how to use the zeros of a polynomial function to construct an equation for a polynomial function when given its graph.

Given the graph of a parabola, which is a quadratic function, the following techniques can be used to construct the corresponding equation. 

• If the vertex can be easily identified, then relate its coordinates to the values $(h, \, k)$. Now write the equation $y=a(x-h)^{2}+k$. Select another point on the parabola, preferably one with integer coordinates, and substitute those values for $x$ and $y$, together with the values for $h$ and $k$, to find the value for $a$.
• If the parabola has zeros at $(p, \, 0)$ and $(q, \, 0)$, then write the equation $y=a(x-p)(x-q)$. Select another point on the parabola, preferably one with integer coordinates, and substitute those values for $x$ and $y$ to find the value for $a$.
• If there is a single zero at $(p, 0)$, then start with the equation $y=a(x-p)^{2}$. Select another point on the parabola, preferably one with integer coordinates, and substitute those values for $x$ and $y$ to find the value for $a$.

Consider the two quadratic graphs shown below. To find the equation for $y=f(x)$, observe that the vertex is located at $(4,-6)$ and that $(0, \, 2)$ is also on the graph. So, substitute $(0,2)$ in the equation $y=a(x-4)^{2}-6$ to find $2=a \cdot 16-6$, so $a=\frac{1}{2}$. Then $y=f(x)=\frac{1}{2}(x-4)^{2}-6$. To find the equation for $y=g(x)$, observe that the zeros are $-5$ and $-1$, so start with the equation $y=a(x+5)(x+1)$. Another point on the graph is $(0,-5)$. Substituting these values into the equation leads to $-5=a(5)(1)$, so $a=-1$. Then $y=g(x)=-(x+5)(x+1)$, which could be written in general form as $y=g(x)=-x^{2}-6 x-5$.

Recall that the graph of a power function is a function of the form $f(x)=k x^{p}$. The end behavior of the power function, together with several ordered pairs on the function, can be used to determine if the function is an even power function or an odd power function, that is, a power function in which the exponent is a nonnegative integer.

Consider the two functions graphed below. The behavior of $s(x)$ suggests it is an odd power function. The points $(1, \, 2),(2, \, 16),(-1, \, -2)$, and $(-2, \, 16)$ appear to be on the graph. So, it appears the equation is $s(x)=2 x^{3}$. In contrast, the behavior of $t(x)$ suggests it is an even power function that has been reflected over the $x$-axis, meaning the coefficient is negative. The points $(2, \, -8)$ and $(-2, \, -8)$ are clearly on the graph and it appears that the points $\left(1,-\frac{1}{2}\right)$ and $\left(-1,-\frac{1}{2}\right)$ are also on the graph. This suggests the equation is $t(x)=-\frac{1}{2} x^{4}$.

If the exponent is not a non-negative integer, then one needs to consider the shape of basic graphs such as $y=x^{1 / 2}, y=x^{1 / 3}$, or $y=x^{-1}$ as a starting point for identifying an equation. Using specific ordered pairs on the graph, together with the basic shapes and transformation concepts, can help determine a feasible equation.

To find the equation of a polynomial function from its graph, start by locating the zeros of the graph. Recall that a polynomial function of degree $n$ has at most $n-1$ turning points and $n$ zeros. These properties can be used to determine the potential degree of the polynomial and the multiplicity of a zero; if $p$ is a zero, then the multiplicity of the zero is the number of times that the factor $(x-p)$ appears in the equation. If a zero has an even multiplicity, then the graph bounces at the zero and does not cross the $x$-axis at that point; if a zero has an odd multiplicity, then the graph crosses the $x$-axis at that point. Use the zeros to write the polynomial in factored form with a leading coefficient of $a$. Find another point on the graph, preferably with integer coordinates, and substitute into the factored form of the equation to find the value of $a$.

Consider the polynomial function shown on the graph. There are two turning points, so the graph is at most degree $3$. The function has zeros at $-1$ and at $5$; the zero at $-1$ bounces off the $x$-axis, so it has an even multiplicity while the zero at 5 crosses the $x$-axis and has an odd multiplicity. Because the degree is at most $3$, the multiplicity at $-1$ must be 2. So, start with the equation $y=h(x)=a(x+1)^{2}(x-5)$. The point (3, -16) appears to be on the graph, so use these coordinates to find $a=\frac{1}{2}$. Then the equation for the polynomial function is $y=h(x)=\frac{1}{2}(x+1)^{2}(x-5)$.

Review material in Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions, Power Functions, Polynomial Functions, and Graphing Polynomial Functions.

5d. apply the intermediate value theorem to determine the approximate location of a zero of a polynomial

• Explain the Intermediate Value Theorem. How can the Intermediate Value Theorem be used to identify intervals in which a zero to a polynomial function must exist?

Recall that a polynomial function is a continuous function, that is, it is a function with no breaks and is defined for all real numbers. The Intermediate Value Theorem is useful for such continuous functions. If $a < b$ and $f(a) \neq f(b)$, then $f(x)$ takes on every value between $f(a)$ and $f(b)$ at least once. For example, suppose a polynomial exists and contains ordered pairs $(5, \, 8)$ and $(9, \, 17)$. Then, for every output value of the function between $8$ and $17$, there is some input between 5 and 9 that results in that output value.

The Intermediate Value Theorem can be used to approximate the location of zeros of a polynomial function. If in some interval $[a, b]$, the values of $f(a)$ and $f(b)$ have opposite signs, meaning that one of these values is above the $x$-axis and the other is below the $x$-axis, then the function must cross the $x$-axis for some value of $c$ in the interval $a < c < b$. This means that $c$ is a zero of the polynomial function because $f(c)=0$. As an example, suppose for some polynomial function that $f(2)=-8$ and $f(5)=10$. Then there is at least one value $c$ in the interval $2 < c < 5$ for which $f(c)=0$; this value of $c$ is a zero of the polynomial.

Review the material in Graphing Polynomial Functions.

5e. evaluate polynomial functions using the remainder and factor theorems

• Contrast the results of the Remainder Theorem and the Factor Theorem when applied to a polynomial function.
• Explain how the Factor Theorem can be used to identify zeros of a polynomial function.

The Remainder Theorem states that if a polynomial function $p(x)$ is divided by the factor $x-k$, the result is the same as evaluating the polynomial for the input value $k$. If $p(k)=0$, then the factor $x-k$ divides the polynomial evenly, meaning that $x-k$ is a factor of the polynomial and $k$ is a zero of the polynomial. This is the special case of the Remainder Theorem known as the Factor Theorem. That is, the Factor Theorem states that if $f(k)=0$ then $(x-k)$ is a factor of the polynomial and $k$ is a zero.

Given the polynomial function $f(x)=4 x^{5}-8 x^{3}+4 x$ and the table of values for the function. Because $f(-1)=0$ and $f(1)=0$, both $x+1$ and $x-1$ are factors of $f(x)$ and $-1$ and 1 are zeros of the function. Because $f(2)=72,2$ is not a zero of the polynomial function; if the polynomial is divided by $x-2$, the remainder is $72$.

Review the material in Identify the x-Intercepts of Polynomial Functions whose Equations are Factorable.

5f. apply the rational zeros theorem, the fundamental theorem of algebra, and the linear factorization theorem to identify zeros of polynomials

• State the Rational Zeros Theorem. How is this theorem used to find zeros of a polynomial?
• Explain how the Linear Factorization Theorem is used to identify zeros of a polynomial.
• What is the Fundamental Theorem of Algebra? How does it help with finding zeros of a polynomial function?
• How are zeros and complex conjugates connected?

The Rational Zeros Theorem, sometimes called the Rational Root Theorem, can be applied to polynomials that have integer coefficients. If the polynomial has rational roots in the form $\frac{p}{q}$, then $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient of the polynomial. This theorem is often used in conjunction with the Factor Theorem from objective 5 e. Evaluate the function for each of the possible rational roots; if the value of the function for any $\frac{p}{q}$ is $0$, then $\frac{p}{q}$ is a root or zero of the polynomial.

For example, consider the polynomial function $f(x)=6 x^{3}+31 x^{2}+3 x-10$. Factors of the leading coefficient are $1,2,3,6$; these are possible values for $q$. Factors of the constant are $1,2,5,10$; these are possible values for $p$. So, possible values of $\frac{p}{q}$ are $\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm 2, \pm \frac{2}{3}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}, \pm 10, \pm \frac{10}{3}$. Given that the polynomial has degree $3$, it has at most 3 zeros. Evaluate the function for each of these values until three roots are found. Evaluating the function for these possible values leads to $f\left(-\frac{2}{3}\right)=f\left(\frac{1}{2}\right)=f(-5)=0$, so the zeros of the polynomial are $-\frac{2}{3}, \frac{1}{2}$, and $-5$.

The Fundamental Theorem of Algebra states that every polynomial has at least one complex zero; so, every polynomial of degree $n > 0$ has $n$ zeros as long as the multiplicity of a zero is considered. Be careful! Not every polynomial has a complex zero of the form $a+b i$ where $b \neq 0$. The real numbers are a subset of the complex numbers, so it might be that the complex zero is of the form $a+b i$ with $b=0$; this is the case when all the zeros of a polynomial are real numbers.

The Linear Factorization Theorem states that a polynomial of degree $n$ has $n$ factors of the form $x-c$ where $c$ is a complex number. If a polynomial has real coefficients and if $a+b i$ with $b \neq 0$ is a zero of the polynomial, the Complex Conjugate Theorem states that $a-b i$ will also be a zero of the polynomial. This theorem has come into use in previous sections when solving quadratic equations; complex roots always occur in conjugate pairs.

These two theorems can be used together to find the equation for a polynomial function if the roots are known. For instance, suppose a fifth degree polynomial has roots $1, \, -4, \, 2+5 i$, and $6$. Because roots must occur in complex conjugate pairs, $2-5 i$ must also be a zero. So, the equation is $f(x)=a(x-1)(x+4)(x-6)(x-[2+5 i])(x-[2-5 i])$. The value of the coefficient $a$ is found by substituting in the coordinates of a specific point on the polynomial function that is not a zero.

Review the material in Identify the x-Intercepts of Polynomial Functions whose Equations are Factorable.

5g. confirm numbers of positive and negative roots of polynomials using Descartes' rule of signs

• Explain how Descartes' Rule of Signs is used to determine the number of positive or negative roots of a polynomial with real coefficients.

Descartes' Rule of Signs provides a test that can be used with other theorems to help in finding the zeros of a polynomial function. Suppose a polynomial $f(x)$ has real coefficients and the terms are written in decreasing order of the exponents. The number of positive zeros equals the number of changes in sign (positive to negative or vice versa) for $f(x)$ or is less than this amount by an even number. The number of negative zeros equals the number of changes in sign (positive to negative or vice versa) for $f(-x)$ or is less than this amount by an even number. Using this information together with the fact that a polynomial of degree $n$ can have at most $n$ zeros helps to know how many positive or negative roots need to be found.

For instance, consider the polynomial $f(x)=3 x^{8}-7 x^{4}+x^{2}-3 x-1$. All the coefficients are real numbers and the exponents are in decreasing order. So, at most there are 8 zeros. There are three sign changes in $f(x)$, so there are 3 or 1 positive roots. Now look at $f(-x)=3 x^{8}-7 x^{4}+x^{2}+3 x-1$; there are three sign changes in $f(-x)$ so there will be 3 or 1 negative roots. The graph of the function below shows 3 negative roots and 1 positive root.

Review the material in Identify the x-Intercepts of Polynomial Functions whose Equations are Factorable.

Unit 5 Vocabulary

This vocabulary list includes terms you will need to know to successfully complete the final exam.

• axis of symmetry of a quadratic function
  
• Complex Conjugate Theorem
• continuous function
• degree of a polynomial
• Descartes' Rule of Signs
• end behavior of a function
• Factor Theorem
• Fundamental Theorem of Algebra
• general form of a quadratic function
• Intermediate Value Theorem
• leading term
• linear Factorization Theorem
• multiplicity of a zero
• polynomial function
• power function
• quadratic function
• Rational Zeros Theorem
• Remainder Theorem
• smooth function
• standard form of a quadratic function
• term of a polynomial
• vertex form of a quadratic function
• vertex of a parabola
• zero of a function