Functions · Book or PDF

Functions 11

Chris Kirkpatrick, Barbara Alldred, Andrew Dmytriw, Shawn Godin, Angelo Lillo, David Pilmer, Susanne Trew, Noel Walker · Grade 11 · Study material generated by AI from this edition. Not affiliated with the publisher.

Chapter 10 · pp. 586–619

Appendix B: Review of Technical Skills

This appendix reviews the technical skills needed for the course. It covers graphing calculators (TI-83 Plus and TI-84) for functions, sequences, series, and finance, and also spreadsheets, The Geometer's Sketchpad, and Fathom. This part covers the calculator skills for graphing, tables, zeros, regression, trigonometry, and sequences.

Getting Ready and Graphing Basic Functions

Preparing the calculator. Old information can spoil a new graph. Leftover list data, an active stat plot, or a stored equation can add stray points or curves. Before you start, clear the lists, turn off all stat plots, delete every equation in the Y= editor, and reset the window so both axes run from -10 to 10. Open the WINDOW screen to check that the reset worked. (p. 587)

Graphing a linear equation. The Y= editor needs y alone on one side. Rewrite the equation as $y = mx + b$ first. For $2x + 3y = 7$ , the result is $y=-\frac{2}{3}x+\frac{7}{3}$ . Type the slope and the intercept inside brackets so the calculator divides correctly. Press GRAPH, then use TRACE and the left and right arrows to read coordinates. If more than one graph is showing, the up and down arrows switch between them. (p. 587)

Evaluating a function. There are two ways to find $y$ for a given $x$ .

Value operation: enter the function in Y=, open the value operation on the CALC menu, and type the $x$ -value, such as -1. The calculator shows $y$ .
Function notation: use the Y-VARS menu to paste Y1 on the home screen, then type the input, as in Y1(37.5). This is handy for inputs that are not whole numbers. (p. 588)

Changing the window. Xmin and Xmax set the domain shown. Ymin and Ymax set the range shown. To display $y=x^2-3x+4$ for $-2\le x\le 5$ and $0\le y\le 14$ , set Xmin = -2, Xmax = 5, Ymin = 0, and Ymax = 14, then press GRAPH. A badly chosen window can hide intercepts or the vertex, so adjust it until the key features appear. (p. 588)

Tables and Differences

Split screens. In the MODE menu you can split the screen. Horiz shows the graph above the equation editor. G-T shows the graph beside a table of values. In G-T mode, TABLE SETUP lets you change where the table starts and the step, for example starting at 0 and going up by 0.5. Seeing both together helps you connect points on the curve to rows in the table. (p. 589)

The TABLE feature. Enter the function in Y=, then set TblStart (the first $x$ -value) and the step size, called $\Delta$ Tbl. Here the Greek letter delta means "change in" $x$ from row to row. For $y=-0.1x^3+2x+3$ , TblStart = -5 and a step of 1 list the values at $x=-5,-4,-3,\dots$ . The arrow keys scroll past either end of the starting range. (p. 589)

First and second differences. Use the STAT lists to build the table.

Put the $x$ -values in L1. For $f(x)=3x^2-4x+1$ , use -2 to 4.
In L2, enter the function written with L1 in place of $x$ .
In L3, apply the DList operation to L2. This gives the first differences.
In L4, apply DList to L3. This gives the second differences.

For a quadratic function, the second differences are constant. Here they are all 6, which equals $2a$ for $a=3$ . (p. 590)

Zeros, Extreme Values, and Inverses

Finding zeros. A zero is an $x$ -intercept. Graph $y=-(x+3)(x-5)$ , then choose the zero operation on the CALC menu. Move the cursor to a point left of the zero and confirm it as the left bound. Move to a point right of the zero and confirm it as the right bound. Confirm once more to accept the guess. The calculator displays the intercept. Each zero needs its own bounds, so repeat the steps for the second one. The zeros here are $x=-3$ and $x=5$ . (pp. 590–591)

Maximum and minimum. A parabola that opens downward has a highest point, so use the maximum operation. A parabola that opens upward has a lowest point, so use the minimum operation. Set a left bound and a right bound on either side of the vertex. For $y=-2x^2-12x+30$ , the calculator gives $(-3, 48)$ . You can check this by hand. The vertex has $x=-\frac{b}{2a}=-\frac{-12}{2(-2)}=-3$ , and then $y=-18+36+30=48$ . (p. 591)

Graphing an inverse with parametric mode. Switch MODE to Par. Both $x$ and $y$ must be written in terms of a parameter $t$ . To graph $y=2-x^2$ with $x\ge 0$ , let $x=t$ and $y=2-t^2$ . To get the inverse, swap the two equations: $x=2-t^2$ and $y=t$ . Keep the original domain, $t\ge 0$ . Swapping $x$ and $y$ reflects the graph in the line $y=x$ , which is how an inverse relates to its function. Set the window to suit the graph. (p. 592)

Data, Regression, and Intersections

Scatter plots. Enter the $x$ -data in L1 and the $y$ -data in L2. For a ball's height over 0 to 6 s, time goes in L1 and height in L2. Turn on Plot 1 with Xlist = L1 and Ylist = L2, then use ZoomStat. ZoomStat sets the window so every point fits. A scatter plot is the first step before you pick a regression model. (p. 593)

Regression. The CALC part of the STAT menu offers LinReg, QuadReg, ExpReg, and SinReg. Choose the one that matches the shape of the scatter plot. You can store the resulting equation in Y1 through the Y-VARS menu, so the curve is drawn with the data. For the baseball data, QuadReg gives about $y=-4.90x^2+29.93x+1.98$ . The value $R^2$ tells how much of the variation in the data the model accounts for. A value near 1 means a good fit. For linear regression, if $r$ does not appear, turn on DiagnosticOn from the catalog and run the regression again. (pp. 593–594)

Points of intersection. Graph both functions and adjust the window so the meeting point is visible. Choose the intersect operation. Confirm the first curve, confirm the second curve, then confirm or move the guess. For $y=5x+4$ and $y=-2x+18$ , the calculator shows $(2, 14)$ . Check: $5(2)+4=14$ and $-2(2)+18=14$ . If there is another intersection, move the cursor close to it and repeat the steps. (p. 594)

Trigonometry in Degrees

Ratios and angles. Set MODE to Degree first. In radian mode, values for degree angles come out wrong. Then $\sin 54^\circ\approx 0.809$ . To find an angle from a ratio, use the inverse keys. For example, $\cos^{-1}(0.6)\approx 53.13^\circ$ . (p. 594)

Graphing a trig function over $0^\circ\le x\le 360^\circ$ . In Degree mode, enter $y=\sin x$ . Set Xmin = 0, Xmax = 360, and Xscl = 90. Since sine stays between -1 and 1, you may set Ymin = -1 and Ymax = 1, but you can skip this and let ZoomFit choose the $y$ -values. ZoomTrig is another option. It always shows $x$ from $-360^\circ$ to $360^\circ$ and $y$ from -4 to 4. (p. 595)

Sequences and Series

Listing terms with seq. On the LIST OPS menu, the seq operation needs five entries: the general term, the variable, the first position, the last position, and the increment. For $t_n=3n-1$ , enter seq(3X - 1, X, 1, 5, 1). The result is 2, 5, 8, 11, 14. The increment is 1 because consecutive positions differ by 1. Other explicit formulas, such as $t_n=n^2$ , work the same way. (p. 596)

Graphing from the general term. Switch MODE to Seq. Enter the formula in u(n) in the Y= editor. In WINDOW, set nMin and nMax. With nMin = 1, the first term is made at $n=1$ . Press GRAPH, then TRACE. The $n$ -value is the position and the $y$ -coordinate is the value of that term. A table with TblStart = 1 and $\Delta$ Tbl = 1 lists the same terms. (pp. 596–597)

Graphing a recursive sequence. The steps are the same, with one extra: the starting value must go in u(nMin). For $t_1=500$ and $t_n=500+0.26t_{n-1}$ , write the rule in u(n) using u(n-1), and set u(nMin) = 500. Without a starting term, the calculator cannot build the later ones. The table then lists the terms in order. (pp. 598–599)

Sum and cumulative sum. The sum of a series is the total of its terms. Use the sum operation on the list that seq produced. The cumulative sum, from cumSum, lists the running total after each term. For $t_n=n^2$ , the first five terms are 1, 4, 9, 16, 25. The cumulative sums are 1, 5, 14, 30, 55. The last of these is the sum of the series. (p. 600)

The TVM Solver: setting it up

The TVM Solver is on the calculator's Finance menu. It holds eight settings. N is the number of payment periods (for a simple annuity, the number of compounding periods). I% is the yearly rate written as a percent, not a decimal. PV is the present value and FV is the future value. PMT is the regular payment. P/Y is payments per year and C/Y is compounding periods per year. The last setting tells the calculator whether payments fall at the BEGIN or the END of each interval. (p. 601)

The sign of each amount matters. Money you pay out is negative (a cash outflow). Money you receive is positive (a cash inflow). To find an unknown, put the cursor on its line and press ALPHA then ENTER (SOLVE). Setting the display to 2 decimal places suits dollar amounts. (p. 601)

Lump sums: future and present value

For a single deposit with no regular payments, set PMT = 0. Let N equal years × periods per year. (p. 602)

Finding FV: $7500 at 8%/a compounded monthly for 9 years. Here N = 9 × 12 = 108, I% = 8, PV = −7500 (you pay it in), P/Y = C/Y = 12. Solving gives FV ≈$ 15 371.48. (p. 602)
Finding PV: Maeve wants $3500 in 5 years at 9%/a compounded quarterly. Enter FV = 3500 (money she receives) and N = 20. Solving gives PV = −$ 2242.86. The negative sign only means she pays that amount now. (p. 602)

Ordinary simple annuities

In an ordinary simple annuity, payments are made at the end of each period, so choose END. (p. 603)

Future value: Celia deposits $1500 at the end of each year for 5 years at 4.5%/a compounded annually. Use PMT = −1500 and PV = 0, with P/Y = C/Y = 1. Solving gives FV =$ 8206.06. (p. 603)
Finding the payment: Mr. Bartolluci wants $150 000 in 15 years at 7%/a compounded monthly. N = 12 × 15 = 180 and FV = 150 000. Solving for PMT gives −$ 473.24 a month. (p. 603)
Present value: A scholarship fund pays out $500 a year for 8 years at 9%/a. Enter PMT = 500 (money received by students) and FV = 0 (the fund ends empty). Solving gives PV = −$ 2767.41, the amount the school must invest now. (p. 604)

Instalment plans and finance charges

The cash price of an item bought on instalments is the present value of the payments plus the down payment. Monica pays $35 a month for 18 months at 16%/a compounded monthly. The present value of the payments is$ 556.82. Adding the $150 down payment gives a cash price of$ 706.82. She actually pays $35 × 18 +$ 150 = $780. The extra amount, about$ 73, is the finance charge. (p. 604)

Finance functions for loans

Once a loan is entered in the TVM Solver, three more functions use its details. Always use a roundvalue of 6, which matches bank practice. (p. 605)

SInt(A, B, roundvalue) adds up the interest paid from payment A to payment B.
SPrn(A, B, roundvalue) adds up the principal paid from payment A to payment B.
bal(x, roundvalue) gives the balance owing after payment x.

Worked loan: Eleanor borrows $18 000 at 14%/a compounded monthly for 5 years. Use N = 60, PV = 18 000 (positive, because she receives the money), FV = 0 and P/Y = C/Y = 12. (p. 605)

Payment: $418.83 a month (the calculator holds 418.8285153 and rounds it).
Total interest, SInt(1, 60): $7129.71. Principal plus interest is$ 25 129.71. Multiplying the rounded payment by 60 gives $25 129.80. The 9-cent gap comes from rounding. (p. 606)
Balance after payment 30: $10 550.27, not$ 9000. Early payments are mostly interest, so the principal drops slowly at first. (p. 606)
The 30th payment splits into $292.33 principal (SPrn(30, 30)) and$ 126.50 interest. (p. 606)

Amortization table on the calculator

To build a table, enter SInt, SPrn and bal in Y1, Y2 and Y3, using X as the payment number. Then set TblStart = 1 and ΔTbl = 1 and open the table. Interest and principal show as negatives because they are outflows. (p. 608) For quick lists, combine seq with bal to list the balances after payments 1 to 7. The balance after 7 payments is $16 486.03. (pp. 607–608)

Scroll to the top and most of each payment is interest. Scroll to the bottom and almost none is. After payment 60 the balance is about $0.00025, essentially zero, so 60 payments do pay off the loan. (p. 608)

Spreadsheets

A cell is named by its column letter and row number, such as A2 or B5. It can hold a label, a number or a formula. A formula starts with =, uses * for multiplication and ^ for powers. Fill Down copies a formula into the selected cells below. (p. 609)

Doubling money: For $1000 at 5%/a compounded quarterly, enter =A2+1 in A3 (quarters), =A3/4 in B3 (years) and =1000*(1.0125)^A3 in C3 (amount). Fill down until the amount passes$ 2000. At 56 quarters (14 years) it is about $2005.03. (p. 609) You can then graph the Time and Amount columns and use the spreadsheet's regression tool to fit a curve. (p. 610)

Amortization table: Buy $4000 of furniture with$ 520 monthly payments at 12%/a compounded monthly, so the monthly rate is 0.01. Each row uses three formulas. Interest = previous balance × 0.01. Principal = payment − interest. New balance = previous balance − principal. Row 1 gives interest $40.00, principal$ 480.00 and balance $3520.00. By payment 9 the balance turns negative, so the final payment must be smaller than$ 520. (p. 611)

The Geometer's Sketchpad

Graphing: Choose Graph > Show Grid, then Plot New Function, and type the expression. For $y=x^2-3x+2$ , type x ^ 2 - 3 * x + 2. Drag the origin to move the axes, or drag the point (1, 0) to change the scale. (pp. 612–613) For $y=2\sin(30x)+3$ , enter the sine from the Functions menu and answer No when asked to switch from degrees. (p. 614)

Triangles and the sine law: Plot three points with the Point tool, select them, and use Construct > Segments. (p. 615) Measure each side with Measure > Length. Measure each angle by selecting three vertices in order, with the angle's vertex in the middle, then Measure > Angle. (p. 616) Use Measure > Calculate to divide each side by the sine of its opposite angle. All three ratios come out equal, which verifies the sine law for that triangle. (p. 617)

Fathom: curve of good fit

Drag a case table into the document, name the attributes and type in the data. Drag a graph in and drop attributes on its axes. (p. 618) Right-click the graph and choose Plot Function. Write an equation with an adjustable parameter such as a. Add a slider for that parameter and move it until the curve matches the points. (p. 619)

Check yourself

In the TVM Solver, why is PV entered as a negative number when you want to find the future value of a deposit?
Why does Eleanor still owe much more than half of her loan after half of her payments?
Before graphing on a TI-83 Plus or TI-84, what should you do to avoid errors from old data, and how do WINDOW settings control what you see?
Describe the steps for finding the zero of a graph, and say which operation you would use for the vertex of a parabola that opens upward.
In a spreadsheet amortization table, write the three formulas for interest, principal and new balance in a row.

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