Update CH 2 to 8

Author: TheElementsMath

01-basics.Rmd

@@ -1,6 +1,6 @@
 # The Basics {#the-basics}
 
-Water engineering calculations depend on reliable algebra. Flow, storage, concentration, pressure, and treatment problems all require careful work with numbers, expressions, equations, and inequalities. This chapter begins with real numbers and arithmetic, then develops sets, exponents, roots, polynomials, equations, lines, rational expressions, and inequalities. Units and physical restrictions remain part of every calculation. A solution is complete only when its algebra is correct, its units are appropriate, and its meaning is reasonable in context.
+Water engineering calculations depend on reliable algebra. Flow, storage, concentration, pressure, and treatment problems all require careful work with numbers, expressions, equations, and inequalities. This chapter begins with real numbers and arithmetic, then develops sets, exponents, roots, polynomials, equations, lines, rational expressions, and inequalities. Units and physical restrictions remain part of every calculation. A solution is complete only when its algebra is correct, its units are appropriate, and its meaning is reasonable in context. 
 
 ::: {.learning-outcomes}
 **Learning outcomes.** 

02-functions.Rmd

@@ -1,6 +1,8 @@
 # Functions {#functions}
 
-Functions provide a precise way to describe how one quantity depends on another. In water engineering, a function may connect time to tank volume, flow to pressure loss, concentration to sensor output, or pipe diameter to cross-sectional area. A function is more than a formula. Its domain states which inputs are permitted, its range states which outputs occur, and its graph shows how the output changes. This chapter follows the progression of the MATH 128 notes from function notation and graphs through rates of change, transformations, combinations, composition, one-to-one and onto behaviour, and inverse functions. Throughout the chapter, units and physical restrictions are treated as part of the model.
+Functions provide a precise way to describe how one quantity depends on another. In water engineering, a function may connect time to tank volume, flow to pressure loss, concentration to sensor output, or pipe diameter to cross-sectional area. A function is more than a formula. Its domain states which inputs are permitted, its range states which outputs occur, and its graph shows how the output changes. 
+
+This chapter progresses from function notation and graphs through rates of change, transformations, combinations, composition, one-to-one and onto behaviour, and inverse functions. Throughout the chapter, units and physical restrictions are treated as part of the model.
 
 ::: {.learning-outcomes}
 **Learning outcomes.** By the end of this chapter, you should be able to:
@@ -68,13 +70,11 @@ The entire expression $x+h$ replaces the original input variable.
 **Domain.** The set of inputs for which a function is defined.
 :::
 
-::: {.definition-box}
-**Range.** The set of outputs a function actually produces.
-:::
+
 
 The algebraic domain is found by excluding operations that are undefined over the real numbers. A denominator cannot equal zero. The radicand of an even root must be nonnegative, and it must be positive when that root appears in a denominator. Logarithmic inputs, introduced later, must be positive. Physical models may add restrictions such as $t\ge0$, $Q\ge0$, or a maximum equipment setting.
 
-Range is often easiest to determine from a graph. The function $g(x)=|x|$ has domain $\mathbb R$ and range $[0,\infty)$. A quadratic opening upward has a minimum output at its vertex. A reciprocal function may exclude a horizontal value. State range restrictions on outputs, not inputs.
+
 
 ::: {.worked-example}
 [**Worked Example: Finding a radical domain**]{.worked-example-title}
@@ -91,6 +91,18 @@ the denominator requires $x(x+3)>0$. A sign chart gives $x<-3$ or $x>0$, so the
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>The denominator requires $x-5>0$, so the domain is $(5,\infty)$.</details>
 :::
 
+
+
+
+::: {.definition-box}
+**Range.** The set of outputs a function actually produces.
+:::
+
+
+Range is often easiest to determine from a graph. The function $g(x)=|x|$ has domain $\mathbb R$ and range $[0,\infty)$. A quadratic opening upward has a minimum output at its vertex. A reciprocal function may exclude a horizontal value. State range restrictions on outputs, not inputs.
+
+
+
 ::: {.worked-example}
 [**Worked Example: Interpreting the range of a storage model**]{.worked-example-title}
 
@@ -131,6 +143,25 @@ Familiar parent graphs make it possible to recognize new functions quickly. Piec
 
 Important parent functions include constant and linear functions, powers such as $x^2$ and $x^3$, roots such as $\sqrt{x}$ and $\sqrt[3]{x}$, reciprocal functions such as $1/x$ and $1/x^2$, and the absolute-value function $|x|$. For each, know the general shape, domain, range, intercepts, symmetry, and asymptotes.
 
+
+
+![](images/basic-function-shapes.png)
+
+
+
+::: {.worked-example}
+[**Worked Example: Recognizing an absolute-value response**]{.worked-example-title}
+
+The function $E(x)=|x-7|$ measures distance from the target 7. Its graph is V-shaped, has vertex $(7,0)$, domain $\mathbb R$, and range $[0,\infty)$.
+:::
+
+::: {.try-it}
+**Try It.** State the vertex and range of $y=|x+3|$.
+<details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>The vertex is $(-3,0)$ and the range is $[0,\infty)$.</details>
+:::
+
+
+
 A piecewise function uses different rules on different parts of its domain. Endpoint symbols determine which rule applies. A solid point represents an included endpoint, while an open point represents an excluded endpoint. Evaluate a boundary input using the rule whose condition includes equality.
 
 ::: {.worked-example}
@@ -148,16 +179,7 @@ Then $P(4)=48$ and $P(5)=60$. The second rule is used at $t=5$ because it includ
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>$P(7)=60+8(7-5)=76$.</details>
 :::
 
-::: {.worked-example}
-[**Worked Example: Recognizing an absolute-value response**]{.worked-example-title}
-
-The function $E(x)=|x-7|$ measures distance from the target 7. Its graph is V-shaped, has vertex $(7,0)$, domain $\mathbb R$, and range $[0,\infty)$.
-:::
 
-::: {.try-it}
-**Try It.** State the vertex and range of $y=|x+3|$.
-<details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>The vertex is $(-3,0)$ and the range is $[0,\infty)$.</details>
-:::
 
 ### Net change, average rate of change, and difference quotients
 
@@ -169,11 +191,7 @@ Between inputs $a$ and $b$, net change is $f(b)-f(a)$ and average rate of change
 
 $$\frac{f(b)-f(a)}{b-a}.$$
 
-Its units are output units per input unit. For inputs $a$ and $a+h$, this becomes the difference quotient
-
-$$\frac{f(a+h)-f(a)}{h},\qquad h\ne0.$$
-
-For a linear function, the average rate of change is constant and equals its slope. For nonlinear functions, it depends on the interval.
+Its units are output units per input unit. 
 
 ::: {.worked-example}
 [**Worked Example: Finding an average inflow rate**]{.worked-example-title}
@@ -188,6 +206,15 @@ $$\frac{57-42}{7-2}=3\ \text{ML/h}.$$
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>$(30-18)/(10-4)=2$ m³/min.</details>
 :::
 
+
+For inputs $a$ and $a+h$, the average value becomes the difference quotient
+
+$$\frac{f(a+h)-f(a)}{h},\qquad h\ne0.$$
+
+For a linear function, the average rate of change is constant and equals its slope. For nonlinear functions, it depends on the interval.
+
+
+
 ::: {.worked-example}
 [**Worked Example: Simplifying a difference quotient**]{.worked-example-title}
 
@@ -226,15 +253,7 @@ Graphs support both qualitative interpretation and numerical problem solving. Th
 
 The solutions of $f(x)=g(x)$ are the $x$-coordinates where their graphs intersect. The inequality $f(x)<g(x)$ holds where the graph of $f$ lies below the graph of $g$. Include intersection points only when equality is part of the comparison.
 
-::: {.definition-box}
-**Increasing function.** A function whose outputs rise as inputs increase on a specified interval.
-:::
 
-::: {.definition-box}
-**Local extremum.** A local maximum or minimum value compared with nearby function values.
-:::
-
-An interval of increase or decrease is described with input values. A turning point may be a local maximum or minimum. A graph can have several local extrema, and a local extremum need not be the greatest or least value on the entire domain.
 
 ::: {.worked-example}
 [**Worked Example: Solving an equation graphically**]{.worked-example-title}
@@ -247,6 +266,22 @@ To solve $x^2-1=8-x^2$, graph $y=x^2-1$ and $y=8-x^2$. Their intersections satis
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>$x=-2$ and $x=2$.</details>
 :::
 
+
+![](images/function-behaviour.png)
+
+::: {.definition-box}
+**Increasing function.** A function whose outputs rise as inputs increase on a specified interval.
+:::
+
+::: {.definition-box}
+**Local extremum.** A local maximum or minimum value compared with nearby function values.
+:::
+
+An interval of increase or decrease is described with input values. A turning point may be a local maximum or minimum. A graph can have several local extrema, and a local extremum need not be the greatest or least value on the entire domain.
+
+
+
+
 ::: {.worked-example}
 [**Worked Example: Describing a quadratic's behaviour**]{.worked-example-title}
 
@@ -262,6 +297,9 @@ For $f(x)=-(x-3)^2+8$, the vertex $(3,8)$ is a maximum. The function increases o
 
 If $c>0$, $f(x)+c$ shifts upward and $f(x)-c$ shifts downward. The graph of $f(x-c)$ shifts right, while $f(x+c)$ shifts left. The graph of $-f(x)$ reflects across the $x$-axis, and $f(-x)$ reflects across the $y$-axis. Multiplying by $a$ changes vertical scale by $|a|$ and reflects vertically when $a<0$. Replacing $x$ by $kx$ changes horizontal scale by $1/|k|$.
 
+
+![](images/function-transformations.png)
+
 Order matters. Work from inside to outside: horizontal scale, horizontal shift, vertical scale or reflection, then vertical shift. Track a small set of key points rather than producing a new table from scratch.
 
 ::: {.worked-example}
@@ -361,12 +399,6 @@ The composition converts time directly to area.
 
 The horizontal line test identifies one-to-one graphs. Algebraically, assume $f(a)=f(b)$ and show that $a=b$. To prove that a function is not one-to-one, one counterexample with two inputs sharing an output is enough.
 
-::: {.definition-box}
-**Onto function.** A function whose range equals its stated codomain.
-:::
-
-Onto depends on the codomain. The rule $f(x)=x^2$ from $\mathbb R$ to $\mathbb R$ is not onto because it produces no negative outputs. The same rule from $\mathbb R$ to $[0,\infty)$ is onto.
-
 ::: {.worked-example}
 [**Worked Example: Testing one-to-one behaviour**]{.worked-example-title}
 
@@ -378,6 +410,16 @@ For $f(x)=5(x-2)^5-7$, suppose $f(a)=f(b)$. Then $(a-2)^5=(b-2)^5$, so $a-2=b-2$
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>No. For example, $h(0)=0$ and $h(3)=0$.</details>
 :::
 
+
+
+::: {.definition-box}
+**Onto function.** A function whose range equals its stated codomain.
+:::
+
+Onto depends on the codomain. The rule $f(x)=x^2$ from $\mathbb R$ to $\mathbb R$ is not onto because it produces no negative outputs. The same rule from $\mathbb R$ to $[0,\infty)$ is onto.
+
+
+
 ### Inverse functions
 
 ::: {.definition-box}

03-polynomial-rational-functions.Rmd

@@ -1,6 +1,10 @@
 # Polynomial and Rational Functions {#polynomial-rational-functions}
 
-Polynomial and rational functions describe curved relationships that linear models cannot capture. Quadratic models represent areas and trajectories, higher-degree polynomials can model responses with several turning points, and rational functions describe ratios whose behaviour may change sharply near excluded inputs. This chapter develops the graphical features emphasized in the MATH 128 notes: vertex form, completing the square, intercepts, local extrema, end behaviour, zeros and multiplicity, asymptotes, holes, and systematic graphing. Water-engineering examples include basin dimensions, pump-response curves, and models whose denominators represent operational limits. Algebraic work is connected continually to graph shape and physical interpretation.
+Polynomial and rational functions describe curved relationships that linear models cannot capture. Quadratic models represent areas and trajectories, higher-degree polynomials can model responses with several turning points, and rational functions describe ratios whose behaviour may change sharply near excluded inputs. 
+
+This chapter develops graphical features of curves: vertex form, completing the square, intercepts, local extrema, end behaviour, zeros and multiplicity, asymptotes, holes, and systematic graphing. 
+
+Water-engineering examples include basin dimensions, pump-response curves, and models whose denominators represent operational limits. Algebraic work is connected continually to graph shape and physical interpretation.
 
 ::: {.learning-outcomes}
 **Learning outcomes.** By the end of this chapter, you should be able to:
@@ -23,11 +27,16 @@ A quadratic function has a parabolic graph whose direction, width, vertex, and i
 
 ### Vertex form and intercepts
 
-In vertex form,
+In vertex form, a quadratic has the form:
+
+$$f(x)=a(x-h)^2+k.$$
+
+The vertex is $(h,k)$. The graph opens upward when $a>0$ and downward when $a<0$. If $a>0$, the vertex gives the minimum value $k$; if $a<0$, it gives the maximum. The magnitude $|a|$ controls vertical stretch or compression. Find $y$-intercepts by setting $x=0$ and $x$-intercepts by solving $f(x)=0$.
+
+
+![](images/vertex-form-parabola.png)
 
-$$f(x)=a(x-h)^2+k,$$
 
-the vertex is $(h,k)$. The graph opens upward when $a>0$ and downward when $a<0$. If $a>0$, the vertex gives the minimum value $k$; if $a<0$, it gives the maximum. The magnitude $|a|$ controls vertical stretch or compression. Find $y$-intercepts by setting $x=0$ and $x$-intercepts by solving $f(x)=0$.
 
 ::: {.worked-example}
 [**Worked Example: Graphing from vertex form**]{.worked-example-title}
@@ -87,7 +96,7 @@ The vertex is $(-5,-18)$.
 
 For $2x^2-12x+5$,
 
-$$2(x^2-6x)+5=2[(x-3)^2-9]+5=2(x-3)^2-13.$$
+$$2(x^2-6x)+5= 2[(x-3)^2-9]+5=2(x-3)^2-13.$$
 
 The vertex is $(3,-13)$.
 :::
@@ -137,6 +146,9 @@ For $P(x)=2x^3-5x^2+x+1$, the leading term is $2x^3$. Therefore, $P(x)\to\infty$
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>As $x\to\infty$, $P(x)\to-\infty$; as $x\to-\infty$, $P(x)\to\infty$.</details>
 :::
 
+
+![](images/polynomial-end-behaviour.png)
+
 ::: {.worked-example}
 [**Worked Example: Bounding the number of turns**]{.worked-example-title}
 
@@ -159,14 +171,20 @@ At a zero of odd multiplicity, a polynomial changes sign and crosses the axis. A
 ::: {.worked-example}
 [**Worked Example: Sketching a factored cubic**]{.worked-example-title}
 
-For $P(x)=5(x+2)(x-1)(x-3)$, the zeros $-2$, 1, and 3 all have multiplicity 1, so the graph crosses at each. The $y$-intercept is 30. Positive cubic end behaviour runs from lower left to upper right.
+For $P(x)=5(x+2)(x-1)(x-3)$, the zeros $-2$, $1$, and $3$ all have multiplicity 1, so the graph crosses at each. The $y$-intercept is 30. Positive cubic end behaviour runs from lower left to upper right.
 :::
 
 ::: {.try-it}
 **Try It.** Describe the intercept behaviour of $x(x-4)^2$.
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>It crosses at $x=0$ and touches without crossing at $x=4$.</details>
 :::
 
+
+
+
+
+
+
 ::: {.worked-example}
 [**Worked Example: Interpreting repeated process thresholds**]{.worked-example-title}
 
@@ -178,12 +196,19 @@ For $R(q)=-q^2(2q+3)(q-1)$, the zero at $q=0$ has multiplicity 2, so the graph t
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>It touches at $x=-1$, crosses at $x=3$, and has positive odd-degree behaviour from lower left to upper right.</details>
 :::
 
+
+
+
+
 ::: {.applied-problem}
 **Applied Problem: Analyzing a treatment-response polynomial.** A dimensionless response is $R(q)=-0.01q(q-20)^2(q-50)$. Identify the zeros and their multiplicities, state where the graph crosses or touches the axis, and describe its end behaviour.
 
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>The zeros are 0 and 50 with multiplicity 1, and 20 with multiplicity 2. The graph crosses at 0 and 50 and touches at 20. The degree is 4 and the leading coefficient is negative, so both ends approach $-\infty$.</details>
 :::
 
+
+![](images/polynomial-zeros-multiplicity.png)
+
 ### Practice Problems
 
 1. Describe the end behaviour of $3x^4-2x+1$. <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>Both ends approach $\infty$.</details>
@@ -226,6 +251,12 @@ the domain excludes $x=1,-2$. There is a hole at $(1,2/3)$ and a vertical asympt
 <details class="answer-dropdown"><summary><strong>Check Your Work</strong></summary>It simplifies to $(x-2)/(x-3)$ with $x\ne-2,3$. There is a hole at $(-2,4/5)$ and a vertical asymptote at $x=3$.</details>
 :::
 
+
+
+![](images/rational-function-hole-asymptote.png)
+
+
+
 ::: {.worked-example}
 [**Worked Example: One-sided behaviour near an operating limit**]{.worked-example-title}