Module 3 (2)

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Baruch College, CUNY**We aren't endorsed by this school

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MATH 4130

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Mathematics

Date

Dec 19, 2024

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Uploaded by BarristerNewt451

Adriana e . Wrasa B q 6. QUADRATIC FUNCTIONS (1) Draw the graph of the quadratic function f(z) = 2(z — 1)? +3. What is its vertex, axis of symmetry, domain, range? Does the parabola open up or down? Do you get maximum vl or minimam value? What s he extremal value, and wheo s it acived? (2) Draw the graph of the quadratic function f(z) = ~2(z + 1)? + 3. What is its vertex, axis of symmtry, domain, rango? Dos the parabola open up or down? Do you got & maximum sl or miimam value? What i tho extromal v, and wheo s it acoved? (3) Write down the standard form of a quadratic function. What is its vertex and its axis of ‘symamotry? When doos the parabola open up or down? When do you got a maximum valuo or & minimum value? What is the extremal value, and where is it achioved? What is its domain and range?

T 30 = (4) Find the coondinates of the vetes, the -inercepts (if an), the -intercept, and dotermine ‘whother the parabola opons up o dowen. Uss this information, and at least owo symmotric points on the parabola to skatch the graph. What is the axis of symmetry, domain, and Fange ofthe function. Fin the minimum or masimum value and dotermine where it occurs. @) =3E-27+4 v o f@)=-3-2" -1 v

T 30 % i) 224120419

i) —82% 4+ 30— 77 T 30

eutl=(z-2p o f@) =5z 22 T 30

T 30 ) (5) Wite an cquation in standard form of each quadratic function whose parabola has the shape of 222 or —32% with the property o Vertexis (1,2), and the parabola opens up. « ertxi (-3 theparbola apens dn Maximum value is 3 which oceurs at x = Minimum value is 4 which occurs at x

MTH 30 60 7. POLYNOMIAL FUNCTIONS AND THEIR GRAPHS (1) What 1s a polynomial function? What 1s 1ts degree, and its leading coefficient? (2) Polynomial functions of degree 2 or higher have graphs that are smooth and continuous. Draw two snu))g)th and continuous graphs which could represent pol_\'r))pmial functions. Draw a gmph} which s not smooth, and draw a graph which is not a}n_ninuom:. (3) Recall the important functions: f(r) = 1, f(z) = =, f(z) = 22, f(z) = 27, f(x) = 7. f(z) =Yz, f(z)=|z|, and f(z) = l Which of these are polynomial functions? Are their T graphs smooth and continuous?

MTH 30 Which of these have discontinuous graphs? Which of these have non-smooth graphs? Which of these are not polynomial and vet have continuous and smooth graphs? 61

(4) Draw tho graghn of 4(x) = 22, f(s) = 2%, f(s) = 2, ami o) -2

MTH 30 62 (5) State in your own words, the Leading Coefficient Test for the graph of a polynomial function. (6) How do vou find the y-intercept of the graph of a function? Explain with an example. (7) When i1s a real number r said to be a zero of a polynomial function? (8) What do the zeroes of a function tell us about the graph of the function? Ilustrate vour answer (use an example). - Y (9) What is the multiplicity of a zero of a polynomial function?

MTH 30 63 (10) Suppose a real number r 1s a zero of a polynomial function with odd multiplicy. What can vou say about the graph of the function at = = r? [llustrate your ams)wer (use an example). L (11) Suppose a real number r 1s a zero of a polynomial function with even multiplicy. What can vou say about the graph of the function at = = r? Illustrate your &Ul.’\i}\}’(‘l‘ (use an example). (12) State the Intermediate Value Theorem for polvnomial functions. (13) Let f(z) = 2 — 22® — 5z + 6. Show that f(0) > 0 and f(2) < 0. What can vou conclude using intermediate value theorem?

T 30 @ (14) Write a parageaph on the turning points of polynomial functions. (15) Using the information above, give a guidcline of how you might draw the graph ofa polyno- mialfuncion. hstrate your ideline wsing /() = ~22%~21% 1076 = ~2(r 1)’ +3)

T 30 s (16) Determine which functions aro polynomial functions. For thoso that ae, dontify the degroe o f@)=2 -4z +5 o) =21 o fle) = —are g o f@) =2~ tr +5VF o f@)=5 (17) Use the Leading Coofficient Tost to determine the end bohaviour of the graph of the poly- nomial function. o J(x) = —Ar" 482 -2 o f(r) = —4s* 4+ 822 - 32 o f(r)=4rt 82032 o flr) =42t 1820 -3

(18) Use the Intermediate Value Theorem to show that each polvnomial has a real zero between the given integers. e f(r) = 22* + 5 — 3; between 0 and 1 e f(xr) =2x* + 22 — 6x; between 1 and 2 MTH 30 66 e f(r) =22 — 322 — 14z + 15; between —2 and —3 (19) Graph the following polynomial functions: o f(r) =3x(x —2)¥(x + 3)® o f(r) = —2r%r +2)¥(zx - 3)?

e f(xr) = —22%(x + 2)*(z — 3)?

« 1) = 32(z (@ + 39 o () =2t - 252 e f@) =2+ -0r -9 T 30 &

e f(x) e (1) e [(1) MTH 30 rd — 12+ 4r —4 -1 4+ T2 - 121 6x® — 12 — ¢