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Solving
Word Problems
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100
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.
100
True or False: The equation for the axis of symmetry for a function f(x)=-ax2+bx+c is
x=b/(2a)
True.
100
How many unique points on a quadratic are needed to find the equation of any given quadratic function?
3 points are needed.
100
Write the Quadratic Formula
100
Factor
x^2-4x-21
(x-7)(x+3)
200
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).
200
True or False: We cannot use the quadratic formula to determine whether a quadratic function has no x-intercepts.
False.
200
The height of a baseball, in meters, at time t, in seconds, is modelled by the function H(t)=-x2+4x+1, how long does it take to reach the ground? (Round to two decimal places)
Set H(t)=0, using technology we obtain t=4.24... and so it takes 4.24s to hit the ground.
200
Simplify
(5k+2)(3-6k)
30k^2+3k+6
200
What is the degree AND leading coefficient of the polynomial below?
4x - 9x2 + 4x3 - 5x4
Degree=4
Leading coefficient=-5
300
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).
300
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).
300
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).
300
Find the vertex of f(x)=-x2+8x-15
(4,1)
300
Write the equation in standard form
y=(x+4)(x+1)(x-1)(x-3)
y=x^4+x^3-13x^2-x+12
400
For 200 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.
400
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.
400
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.
400
Simplify
(3x^3y+9x+3y^2+7y)+(8x^3y+9y^2)
11x^3y+12y^2+9x+7y
400
Find the Vertex:
f(x)=-x2-4x
(-2,4)
500
The quadratic, f(x), can be written as f(x)=ax2+bx+10 in standard form and f(x)=(2x-2)(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)+(-2)(x)+(-2)(-5)
=2x2-10x-2x+10
so a=2
so b=-12
500
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.
500
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.
500
Factor
4x2-22x+30
2(2x-5)(x-3)
500
Write the equation in factored form:
y=(x+4)(x+1)(x-1)(x-3)