Name that form
What is the definition of a function?
A function is a relation between variables, x and y, where every x-value results in only one y-value.
True or False: The equation for the axis of symmetry for a function f(x)=-ax2+bx+c is
x=b/(2a)
True.
True or False: The maximum area of a rectangle given any fixed perimeter will be a square.
True.
True or False: The point (-2,-12) is on the quadratic f(x)=-3x2-2x+4.
False:
f(-2)=-3(-2)2-2(-2)+4
=-3(4)+4+4
=-12+8
=-4
not equal to -12.
True or False: We cannot use the quadratic formula to determine whether a quadratic function has no x-intercepts.
False.
Rewrite the quadratic
y=-2(x-1/2)2-(1/4)
in standard form.
In standard form it is given by, y=-2x2-x+(1/4).
What is the coordinate of the y-intercept of the quadratic
y=(x+1)2+2
Set x=0 to find the y-intercept, we then have,
y=(0+1)2+2
=3
So, the coordinate of the y-intercept is (0,3).
100m of fencing is available to build a rectangular enclosure where one side is a brick wall with fixed length. Write an expression for the area of the enclosure with width w.
The area of the enclosure is given by A(w)=w(100-2w).
True or False: The functions,
f(x)=x2-3
and,
g(x)=(1/3)(x+2)2+1
intersect at the point (-2,1). For a bonus 100 points, do they intersect at any other points? – if so, where?
True:
f(-2)=(-2)2-3=4-3=1
g(-2)=(1/3)(-2+2)2+1=0+1=1
They also intersect at the point (1,-2).
How many unique points on a quadratic are needed to find the equation of any given quadratic function?
3 points are needed.
For 100 points, what form is the quadratic
y=-x2+3x+4
in? For another 100 points, rewrite it in factored form.
It is in standard form, in factored form it is given by y=-(x-1)(x+3).
Determine the equation of the axis of symmetry of the quadratic,
f(x)=-5x2-(3/4)x+7
We have,
x=-b/(2a)
=-(-3/4)/(2(-5))
=3/(-10(4))
=-3/40
So, the axis of symmetry occurs at x=-3/40.
The height of a baseball, in metres, at time t, in seconds, is modelled by the function H(t)=-x2+4x+1, how long does it take to reach the ground? (3 sig digs)
Set H(t)=0, using technology we obtain t=4.24... and so it takes 4.24s to hit the ground.
How many unique point(s) of intersection do f(x)=2x2+4 and g(x)=3x2-2x+5 have?
f(x)=g(x)
2x2+4=3x2-2x+5
x2-2x+1=0
(x-1)2=0
x=1
So, f(x) and g(x) have 1 unique point of intersection.
True or False: The functions f(x)=x2+5x+12 and g(x)=-x2-5x+13 have the same axis of symmetry.
True:
for f(x),
x=-b/(2a)=-5/(2(1))=-5/2
and for g(x),
x=-b/(2a)=-(-5)/(2(-1))=-5/2
For 100 points, what form is the function
f(x)=-0.4(x-1)(x+1)
in? For another 200 points, what is the axis of symmetry of f(x)?
f(x) is in factored form. Averaging the x-intercepts, we get the axis of symmetry at x=0.
What are the x-intercept(s) of the quadratic function y=x2+2x+1.
Factoring we obtain
y=(x+1)(x+1)
So we have the x-intercept at x=1. We can also us the GDC by graphing or using poly solver.
Profit from ticket sales for a venue is modelled by P(t)=-(1/10)x2+3x+15. How many tickets should be sold to maximize profit?
We want to find the axis of symmetry, so,
x=-b/(2a)=-3/(2(-1/10))=(3*10)/2=15
So, to maximize profit, 15 tickets should be sold.
If the functions f(x)=-x2+8x+c and g(x)=-(4-x)2 intersect at infinitely many points, determine the value of c.
This implies f(x)=g(x), so,
-x2+8x+c=-(4-x)2
-x2+8x+c=-(16+x2-8x)
So,
c=-16
If the axis of symmetry of f(x)=ax2+3x+2 occurs at x=-2, what is the value of a?
The axis of symmetry of f(x) occurs at,
x=-2=-b/(2a)=-3/(2a)
so -2=-3/(2a)
and, a=3/4.
The quadratic, f(x), can be written as f(x)=ax2+bx+10 in standard form and f(x)=(2x-n)(x+5) in factored form. For 200 points, what is the value of a? For another 300, what is the value of b?
f(x)=(2x)(x)+(2x)(5)+(-n)(x)+(-n)(5)
=2x2+10x-nx-5n
so a=2
and since -5n=10, n=-2
so b=10-n=10-(-2)=12.
If 2(3x-2)(x+1)=2 and the domain is restricted to positive real numbers, what is x?
2(3x2+3x-2x-2)=2
6x2+2x-4=2
6x2+2x-2=0
x=1/2.
A picture frame has a thickness of t inches. The picture has dimensions 5 inches by 10 inches. Write an expression for the area taken up by the frame.
For a bonus 200 points, find t if the maximum area taken up by the frame (with the picture) mounted on the wall is 500 in2.
A=(5+2t)(10+2t).
t=7.5 in.
The functions f(x)=x2+bx+6 and g(x)=-2x2-8x+c intersect exactly once at their vertices, for 400 points, determine the value of b. For another 200 points (100 bonus!), determine the value of c.
b=4, c=6.
200 point bonus! A function, f(x)=ax2+c, passes through the point (s,10) and the point (10,-s) for some real number s. If the vertex of f(x) occurs at (0,1), find the value of s.
Hint: symmetry
s=-10.