Find the two numbers whose sum is 30 and the product is maximum.
a.
15, 15
b.
12, 18
c.
14, 16
d.
13, 17
Solution:
Let x represents one number.
Let 30 - x represents the other number. [Sum is 30.]
Let y represents the product of the two numbers.
y = x(30 - x) [Express as an equation.]
y = - x^{2} + 30x [Multiply.]
y = - x^{2} + 30x is a quadratic with a = - 1 and b = 30.
[Compare with y = ax^{2} + bx + c.]
a = -1 < 0. So, the parabola opens downward with its vertex being the maximum point.
This maximum point occurs when x = - b2a.
x = - 30 / 2(-1) = 15 [Substitute and simplify.]
If x = 15, then 30 - x = 30 - 15 = 15.
So, the numbers are 15 and 15.
Correct answer : (1)
2.
Find two numbers whose difference is 12 and their product is minimum.
a.
4, - 8
b.
6, - 6
c.
- 9, 3
d.
7, - 5
Solution:
Let x represents one number.
Let 12 + x represents the other number. [Since the difference is 12.]
Let y represents the product of the two numbers.
y = x(12 + x) [Express as an equation.]
y = x^{2} + 12x [Multiply.]
y = x^{2} + 12x is a quadratic with a = 1 and b = 12. [Compare with y = ax^{2} + bx + c.]
a = 1 > 0. So, the parabola opens upward with its vertex being the minimum point.
This minimum point occurs when x = - b2a.
x = - 12 / 2(1)= - 6 [Substitute and simplify.]
If x = - 6, then 12 + x = 12 + (- 6) = 6.
So, the numbers are 6 and - 6.
Correct answer : (2)
3.
Check if the function $y$ = - 4$x$^{2} + 3 has a maximum or minimum value. Find that value.
a.
Maximum; 3
b.
Minimum; 3
c.
Maximum; - 3
d.
Minimum; - 3
Solution:
y = - 4x^{2} + 3 is a quadratic function with a = - 4, b = 0 and c = 3. [Compare with y = ax^{2} + bx + c.]
a = - 4 < 0. So, the parabola opens downward with its vertex being the maximum point.
This maximum point occurs when x = - b2a.
x = - 0 / 2(-4)= 0 [Substitute and simplify.]
Substitute this x - value in the given quadratic function to find the corresponding y - value.
y = - 4x^{2} + 3
= - (4)(0)^{2} + 3 = 3
For a quadratic function, the y - coordinate of the vertex is the maximum or
minimum value of the function.
So, the given function has a maximum value 3.
Correct answer : (1)
4.
Sketch the graph of $y$ = $x$^{2} + 10$x$ + 25.
a.
Graph-4
b.
Graph-1
c.
Graph-2
d.
Graph-3
Solution:
y = x^{2} + 10x + 25
a = 1, b = 10 and c = 25 [Compare with y = ax^{2} + bx + c.]
a = 1 > 0. So, the graph opens upward.
x - coordinate of the vertex = - b2a = - (10) / 2(1) = - 10 / 2= - 5
y - coordinate of the vertex = f( - 5) = (- 5)^{2} + 10(- 5) + 25
= 25 - 50 + 25 = 0
So, the vertex is the point (- 5, 0) and the axis of symmetry is the line x = - 5.
x^{2} + 10x + 25 = 0 [Substitute the values.]
x = - 5 (twice) [Factor and simplify.]
c = 25, so, y - intercept = 25.
The graph just touches the x - axis at the point (- 5, 0).
The y - intercept (0, 25) is 5 units to the right of the axis of symmetry. So, its image point would be 5 units to the left of the axis. It is: (- 10, 25).
Graph the axis of symmetry. Plot all the points found and connect them with a smooth curve to get the graph of the given function.
Correct answer : (3)
5.
Find the equation of the axis of symmetry, the coordinates of the vertex, and the $x$ - and $y$ - intercepts for the function $y$ = - $x$^{2} + 6$x$.
y = - x^{2} + 6x is a quadratic with a = - 1, b = 6 and c = 0.
[Compare with y = ax^{2} + bx + c.]
The equation of the axis of symmetry is given by: x = - b2a
= - 6 / 2(-1)= 3 [Substitute and simplify.]
So, the equation of the axis of symmetry is x = 3.
The x - coordinate of the vertex of a quadratic function is given by:
x = - b2a = 3
Substitute the x - coordinate of the vertex in the given quadratic function to find the corresponding y - coordinate.
y = - x^{2} + 6x
= - (3)^{2} + 6(3) = 9
So, the vertex of the given quadratic function is (3, 9).
Substitute y = 0 in the given quadratic function to find the x - intercept.
- x^{2} + 6x = 0
- x (x - 6) = 0 [Factor.]
Therefore, x = 0 or x = 6. [Solve for x.]
c = 0, so, y - intercept = 0.
Correct answer : (2)
6.
Find the equation of the axis of symmetry, the coordinates of the vertex, and the $x$ - and $y$ - intercepts for the function $y$ = $x$^{2} + 12$x$ + 36.
a.
$x$ = 6; (6, 0); - 6; 36
b.
$x$ = - 6; ( - 6, 72); - 6; 36
c.
$x$ = - 6; ( - 6, 0); 6; 36
d.
$x$ = - 6; ( - 6, 0); - 6; 36
Solution:
y = x^{2} + 12x + 36 is a quadratic with a = 1, b = 12 and c = 36. [Compare with y = ax^{2} + bx + c.]
The equation of the axis of symmetry is given by x = - b2a
= - 12 / 2(1)= - 6 [Substitute and Simplify.]
So, the equation of the axis of symmetry is x = - 6.
The x - coordinate of the vertex of a quadratic function is given by
x = - b2a = - 6
Substitute the x - coordinate of the vertex in the given quadratic function to find the corresponding y - coordinate.
y = x^{2} + 12x + 36
= (- 6)^{2} + 12( - 6) + 36 = 0
So, the vertex of the given quadratic function is (- 6, 0).
Substitute y = 0 in the given quadratic function to find the x - intercept.
x^{2} + 12x + 36 = 0
(x + 6)^{2} = 0 [Factor.]
Therefore, x = - 6 (twice). [Solve for x.]
c = 36, so, y - intercept = 36.
Correct answer : (4)
7.
For what values of $a$ and $b$, the function $y$ = $a$$x$^{2} + $\mathrm{bx}$ + 6 will have its vertex at (- 6, -138)?
a.
$a$ = 4, $b$ = - 48
b.
$a$ = 4, $b$ = 48
c.
$a$ = - 4, $b$ = 48
d.
$a$ = - 4, $b$ = - 48
Solution:
y = ax^{2} + bx + 6 is a quadratic with a = a, b = b and c = 6. [Compare with y = ax^{2} + bx + c.]
The x - coordinate of the vertex of a quadratic function is given by:
x = - b2a
- b2a = - 6
b = 12a [Simplify.]
That is: 12a - b = 0 - - - - Equation (1) [Express as an equation.]
Substitute the vertex (- 6, -138) in the given equation: y = ax^{2} + bx + 6.
-138 = a(- 6)^{2} + b(- 6) + 6
36a - 6b = -144 [Simplify.]
6a - b = -24 - - - - Equation (2) [Divide throughout by 6.]
Solve equations (1) and (2) to get the values of a and b.
We get a = 4 and b = 48.
Correct answer : (2)
8.
Find the equation of the axis of symmetry, the coordinates of the vertex, and the $x$ - and $y$ - intercepts for the function $y$ = $x$^{2} + 2$x$ - 80.
y = x^{2} + 2x - 80 is a quadratic with a = 1, b = 2 and c = - 80. [Compare with y = ax^{2} + bx + c.]
The equation of the axis of symmetry is given by: x = - b2a
= - 2 / 2(1)= - 1 [Substitute and simplify.]
So, the equation of the axis of symmetry is x = - 1.
The x - coordinate of the vertex of a quadratic function is given by:
x = - b2a = - 1
Substitute the x - coordinate of the vertex in the given quadratic function to find the corresponding y - coordinate.
y = x^{2} + 2x - 80
= (- 1)^{2} + 2( - 1) - 80 = -81
So, the vertex of the given quadratic function is (- 1, -81).
Substitute y = 0 in the given quadratic function to find the x - intercept.
x^{2} + 2x - 80 = 0
(x + 10)(x - 8) = 0 [Factor.]
Therefore, x = - 10, 8. [Solve for x.]
c = -80, so, y - intercept = - 80.
Correct answer : (2)
9.
Find the equation of the axis of symmetry, the coordinates of the vertex, and the $x$ - and $y$ - intercepts for the function $y$ = $\frac{1}{4}{x}^{2}-\frac{7}{4}x$ + 3.
a.
$x$ = ($\frac{-1}{16}$); x - intercept = - 4 or 3; y - intercept = 12
b.
$x$ = $\frac{7}{2}$ ; ($\frac{7}{2}$ , $\frac{-1}{16}$ ); x - intercept = 4 or 3; y - intercept = 12
c.
$x$ = - $\frac{7}{8}$; x - intercept = 4 or 3; y - intercept = 12
d.
None of the above
Solution:
y = 14x2-74x + 3 is a quadratic with a = 1 / 4, b = - 7 / 4 and c = 3.
[Compare with y = ax^{2} + bx +c.]
The equation of the axis of symmetry is given by: x = - b2a
= - -742×14 = 7 / 2 [Substitute and simplify.]
So, the equation of the axis of symmetry is x = 7 / 2 .
The x - coordinate of the vertex of a quadratic function is given by:
x = - b2a = 7 / 2.
Substitute the x - coordinate of the vertex in the given quadratic function to find the corresponding y - coordinate.
y = 1 / 4x^{2} - 7 / 4x + 3
= 1 / 4(7 / 2)^{2} - 7 / 4(7 / 2) + 3 = -1 / 16.
So, the vertex of the given quadratic function is (7 / 2, -1 / 16).
Substitute y = 0 in the given quadratic function to find the x - intercept.
1 / 4x^{2} - 7 / 4x + 3 = 0
x^{2} - 7x + 12 = 0 [Multiply throughout by 4.]
(x - 4)(x - 3) = 0 [Factor.]
Therefore, x = 4, 3. [Solve for x.]
c = 12, so, y - intercept = 12.
Correct answer : (2)
10.
Check if the function $y$ = - $x$^{2} + 12$x$ + 8 has a maximum or minimum value. Find that value.
a.
Maximum; 44
b.
Maximum; 46
c.
Minimum; 49
d.
Minimum; 44
Solution:
y = - x^{2} + 12x + 8 is a quadratic with a = - 1, b = 12 and c = 8.
[Compare with y = ax^{2} + bx + c.]
a = - 1 < 0. So the parabola opens downward with its vertex being the maximum point.
This maximum point occurs when x = - b2a.
x = - 122(-1) = 6 [Substitute and simplify.]
Substitute this x - value in the given quadratic function to find the corresponding y - value.
y = - x^{2} + 12x + 8
= - (6)^{2} + 12(6) + 8 = 44
For a quadratic function, the y - coordinate of the vertex is the maximum or
minimum value of the function.