Compare the function with the parent function. Without graphing, what are the vertex, axis of symmetry, and transformations of the given function?
y= |10x-2|-7
A. (1/5,-7); x = 1/5; translated to the right 1/5 unit and down 7 units.
B. (1/5,-7); x = 1/5; translated to the left 1/5 unit and up 7 units.
C. (1/5,7); x = 1/5; translated to the left 1/5 unit and down 7 units.
D. (1/5,7); x = 1/5; translated to the right 1/5 unit and down 7 units.
Every year in Delaware there is a contest where people create cannons and catapults designed to launch pumpkins as far in the air as possible. The equations y = 12 + 105x - 16x^2 can be used to represent the height, y, of a launched pumpkin, where x is the time in seconds that the pumpkin has been in the air. What is the maximum height that the pumpkin reaches? How many seconds have passed when the pumpkin reaches? (hint if the pumpkin hits the ground its height is 0 feet.)
To find the vertex, axis of symmetry, and transformations of the given function, y = |10x-2|-7, we can start by looking at the parent function, which is y = |x|.
The vertex of the parent function, y = |x|, is at (0, 0) because the absolute value of any number is always non-negative. The axis of symmetry is the y-axis because the function is symmetric with respect to it.
Now let's consider the given function, y = |10x-2|-7. We can see that there are two transformations applied to the parent function.
1. Horizontal translation:
The expression inside the absolute value, 10x-2, represents a horizontal translation. The constant term (-2) is subtracted from x, which means the function has been translated to the right by 2/10 or 1/5 unit.
2. Vertical translation:
The constant term (-7) outside the absolute value represents a vertical translation. The function has been shifted downward by 7 units.
As a result, we now have enough information to compare the given function with the parent function:
- The vertex of the given function is (1/5, -7) because of the horizontal translation to the right by 1/5 unit and the vertical translation downward by 7 units.
- The axis of symmetry of the given function is x = 1/5 because the equation represents a vertical line passing through the vertex.
- The answer is option A: (1/5, -7); x = 1/5; translated to the right 1/5 unit and down 7 units.
Now let's move on to the second question about the pumpkin launch.
The equation provided, y = 12 + 105x - 16x^2, represents the height, y, of a pumpkin launched over time, x.
To find the maximum height that the pumpkin reaches, we can examine the equation and recognize that it is a quadratic function in the form of y = ax^2 + bx + c, where a = -16, b = 105, and c = 12.
Since the coefficient of the x^2 term (a) is negative, we know that the parabola opens downward, indicating a maximum. The maximum height occurs at the vertex of the quadratic equation.
To find the x-value of the vertex (the time it takes for the pumpkin to reach its maximum height), we can use the formula x = -b/2a. Plugging in the values, we get:
x = -105 / 2(-16)
x = -105 / -32.
Simplifying, we find x ≈ 1.43 seconds.
Now, to find the maximum height, we substitute this x-value back into the equation:
y = 12 + 105(1.43) - 16(1.43)^2.
y ≈ 181.23 feet.
Therefore, the maximum height that the pumpkin reaches is approximately 181.23 feet, and it takes approximately 1.43 seconds for the pumpkin to reach that height.
For the given function y = |10x-2|-7:
Vertex: To find the vertex of the given function, we need to determine the x-coordinate of the vertex by setting the expression inside the absolute value equal to zero and solving for x:
10x - 2 = 0
10x = 2
x = 2/10
x = 1/5
Plugging this x-value into the function, we can find the y-coordinate of the vertex:
y = |10(1/5)-2|-7
y = |-2/5-2|-7
y = |-12/5|-7
y = (12/5)-7
y = (12/5)-(35/5)
y = -23/5
y ≈ -4.6
So, the vertex is approximately (1/5, -4.6).
Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. Since the vertex in this case is (1/5, -4.6), the axis of symmetry is x = 1/5.
Transformations: Comparing the given function to the parent function y = |x|, we can see that the given function is obtained by performing the following transformations:
1. Translated horizontally to the right 1/5 unit.
2. Translated vertically down 7 units.
Therefore, the correct answer is A. (1/5, -7); x = 1/5; translated to the right 1/5 unit and down 7 units.
Now, let's work on the other question:
The given equation is y = 12 + 105x - 16x^2. This is a quadratic function in the form of y = ax^2 + bx + c, where a = -16, b = 105, and c = 12.
To find the maximum height that the pumpkin reaches, we can use the vertex form of a quadratic function. The vertex form is given by y = a(x-h)^2 + k, where (h, k) represents the vertex.
In this case, the vertex for the given function y = 12 + 105x - 16x^2 can be found by using the formula x = -b/(2a). Plugging in the values, we get:
x = -105/(2*-16)
x = 105/32
To find the y-coordinate of the vertex, we substitute this x-value into the original function:
y = 12 + 105(105/32) - 16(105/32)^2
y ≈ 5296.41
So, the maximum height that the pumpkin reaches is approximately 5296.41 feet.
Next, to find the number of seconds it takes for the pumpkin to reach this height, we need to find the time when y = 0 (the height when the pumpkin hits the ground). In this case, the y-coordinate is 0. We can solve for x using the quadratic formula, which is given by:
x = (-b ± √(b^2-4ac))/(2a)
Plugging in the values, we get:
x = (-105 ± √(105^2-4*-16*12))/(2*-16)
x = (-105 ± √(11025+768))/(2*-16)
x = (-105 ± √11893)/(2*-16)
Since the time cannot be negative, we only consider the positive value:
x ≈ 8.12
So, it takes approximately 8.12 seconds for the pumpkin to reach the ground.
Therefore, the maximum height that the pumpkin reaches is approximately 5296.41 feet, and it takes approximately 8.12 seconds for the pumpkin to reach the ground.
|10x-2| = |10(x - 1/5)|-7
so, what do you think?
the vertex of ax^2+bx+c is at x = -b/2a