The function h(t) = −5t2 + 20t shown in the graph models the curvature of a satellite dish: graph of parabola starting at zero comma zero, rising from the left to 2 comma 20, and falling to the right, ending at 4 comma zero What is the domain of h(t)? A: x ≥ 0 B: 0 ≤ x ≤ 4 C: 0 ≤ x ≤ 20 D: All real numbersPierce removes the plug from a trough to drain the water. The volume, in gallons, in the trough after it has been unplugged can be modeled by the expression 10x2 −17x + 3, where x is the time in minutes. Choose the appropriate form of the expression that would reveal the time in minutes when the trough is empty.A: 10(x − 3)2 − 1B: 10(x − 1)2 − 3C: (5x − 1)(2x − 3)D: 10(0)2 − 17(0) + 3A farmer is tracking the number of soybeans his land is yielding each year. He finds that the function f(x) = −x2 + 20x + 100 models the crops in pounds per acre over x years. Find and interpret the average rate of change from year 10 to year 20.A: The crop yield increased by 200 pounds per acre from year 10 to year 20.B: The crop yield increased by 100 pounds per acre from year 10 to year 20.C: The crop yield decreased by 10 pounds per acre from year 10 to year 20.D: The crop yield decreased by 0.1 pounds per acre from year 10 to year 20.A function is shown: f(x) = 4x2 − 1.Choose the equivalent function that best shows the x-intercepts on the graph.A: f(x) = (4x + 1)(4x − 1)B: f(x) = (2x + 1)(2x − 1)C: f(x) = 4(x2 + 1)D: f(x) = 2(x2 − 1)A particular company's net sales, in billions, from 2008 to 2018 can be modeled by the expression t2 + 14t + 85, where t is the number of years since the end of 2008. What does the constant term of the expression represent in terms of the context?A: The company earned 85 billion dollars in 2008.B: The company earned 14 billion dollars in 2008.C: The company earned 85 billion dollars from 2008 to 2018.D: The company earned 14 billion dollars from 2008 to 2018.What are the x-intercepts of the parabola?graph of parabola falling from the left, passing through 2 comma 3 to about 4 comma negative 1, and rising to the right, passing through 6 comma 3A: (0, 3) and (0, 5)B: (0, 4) and (0, 5)C: (3, 0) and (5, 0)D: (4, 0) and (5, 0)

Answer:Part 1) Option B: 0 ≤ x ≤ 4Part 2) Option C: (5x − 1)(2x − 3)Part 3) Option C: The crop yield decreased by 10 pounds per acre from year 10 to year 20.Part 4) Option B: f(x) = (2x + 1)(2x − 1)Part 5) Option A: The company earned 85 billion dollars in 2008Part 6) Option C: (3, 0) and (5, 0)Step-by-step explanation:Part 1) we have[tex]h(t)=-5t^{2}+20t[/tex]This is a vertical parabola open downwardThe vertex is a maximumThe x-intercepts are the points (0,0) and (4,0)The vertex is the point (2,20) ---> The x-coordinate of the vertex is the midpoint of the x-interceptssoThe domain is the interval ----> [0,4][tex]0\leq t\leq 4[/tex]The range is the interval ----> [0,20][tex]0\leq h(t)\leq 20[/tex]Part 2) we have [tex]V=10x^{2} -17x+13[/tex]wherex is the time in minutesV is the volume in gallonswe know thatWhen the trough is empty, the volume is equal to zerosoFor V=0[tex]10x^{2} -17x+3=0[/tex]The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to [tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex] in this problem we have [tex]10x^{2} -17x+3=0[/tex]so [tex]a=10\\b=-17\\c=3[/tex] substitute in the formula [tex]x=\frac{-(-17)(+/-)\sqrt{-17^{2}-4(10)(3)}} {2(10)}[/tex] [tex]x=\frac{17(+/-)\sqrt{169}} {20}[/tex] [tex]x=\frac{17(+/-)13} {20}[/tex] [tex]x=\frac{17(+)13} {20}=\frac{3}{2}[/tex] [tex]x=\frac{17(-)13} {20}=\frac{1}{5}[/tex] so[tex]10x^{2} -17x+3=10(x-\frac{3}{2})(x-\frac{1}{5})[/tex]simplify[tex]10(x-\frac{3}{2})(x-\frac{1}{5})=(2x-3)(5x-1)[/tex]Part 3) we have[tex]f(x)=-x^{2}+20x+100[/tex]we know thatTo find the average rate of change, we divide the change in the output value by the change in the input value the average rate of change is equal to [tex]\frac{f(b)-f(a)}{b-a}[/tex] In this problem we have [tex]f(a)=f(10)=-(10)^{2}+20(10)+100=200[/tex]  [tex]f(b)=f(20)=-(20)^{2}+20(20)+100=100[/tex]  [tex]a=10[/tex] [tex]b=20[/tex] Substitute[tex]\frac{100-200}{20-10}[/tex] [tex]-\frac{100}{10}=-10[/tex]  ----> is a decreasing functionthereforeThe crop yield decreased by 10 pounds per acre from year 10 to year 20.Part 4) we have[tex]f(x)=4x^{2} -1[/tex]Find out the x-interceptsRemember that the x-intercepts are the values of x when the value of the function is equal to zerosoFor f(x)=0[tex]4x^{2} -1=0[/tex]Solve for x[tex]4x^{2}=1[/tex][tex]x^{2}=\frac{1}{4}[/tex]take square root both sides[tex]x=(+/-)\frac{1}{2}[/tex]so[tex]f(x)=4x^{2} -1=4(x-\frac{1}{2})(x+\frac{1}{2})=(2x-1)(2x+1)[/tex]Part 5) we have[tex]t^{2}+14t+85[/tex]Remember thatThe y-intercept of a function is the value of the function (output value) when the value of x (input value) is equal to zeroIn this problemThe expression represent a company's net sales, in billions, from 2008 to 2018 t is the number of years since the end of 2008soFor t=0 -----> represent the end of 2008 [tex](0)^{2}+14(0)+85=85\ billion\ dollars[/tex]thereforeThe company earned 85 billion dollars in 2008.Part 6) What are the x-intercepts of the parabola?we have the points (2,3),(4,-1),(6,3)This is a vertical parabola open upwardThe vertex represent a minimumIn this problem the vertex is the point (4,-1)The equation of a vertical parabola in vertex form is equal to[tex]y=a(x-h)^2+k[/tex]wherea is a coefficient(h,k) is the vertexwe have(h,k)=(4,-1)substitute[tex]y=a(x-4)^2-1[/tex]take the point (2,3) and substitute in the quadratic equation to solve for aFor x=2, y=3[tex]3=a(2-4)^2-1[/tex][tex]3=4a-1[/tex][tex]4a=4[/tex][tex]a=1[/tex]The quadratic equation is[tex]y=(x-4)^2-1[/tex]Find out the x-interceptsRemember that the x-intercepts are the values of x when the value of the function is equal to zerosoFor y=0[tex](x-4)^2-1=0[/tex]Solve for x[tex](x-4)^2=1[/tex]take square root both sides[tex](x-4)=(+/-)1[/tex][tex]x=4(+/-)1[/tex][tex]x=4+1=5\\x=4-1=3[/tex]thereforeThe x-intercepts are the points (3,0) and (5,0)