Find the vertex and axis of symmetry of y = -2x^2 + 4x + 1. Show all of your work and justify your reasoning.
Note: The ^ symbol means to the power.
For a quadratic equation :
y = a x ^ 2 + b x + c
the vertex ( x = h , y = k ) is found by computing :
x = h = – b / 2 a
and then evaluating y at h to find k.
In this case :
a = - 2
b = 4
h = - b / 2 a =
- 4 / [ 2 * ( - 2 ) ] =
- 4 / - 4
h = 1
The x - coordinate of the vertex :
x = 1
k = y = - 2 x ^ 2 + 4 x + 1
k = - 2 * 1 ^ 2 + 4 * 1 + 1
k = - 2 * 1 + 4 + 1
k = - 2 + 4 + 1
k = 3
y = 3
The y - coordinate of the vertex :
y = 3
The axis of symmetry is a vertical line through the vertex, so its equation is :
x = 3
( For all value of y , x = 1 )
Tipefeler:
The axis of symmetry is a vertical line through the vertex, so its equation is :
x = 1
( For all value of y , x = 1 )
or ... just complete the square
y = -2(x^2 -2x + ... ) + 1
= -2(x^2 - 2x + 1 - 1) + 1
= -2(x- 1)^2 + 3
all your requests can be read off that.
The drag force F (in pounds) of water on a swimmer can be modeled by F = 1.35s^2 where s is the swimmer’s speed ( in miles per hour). At what speed is the force minimized? **Note** ^ means "to the power of."
To find the vertex and axis of symmetry of a quadratic equation in the form y = ax^2 + bx + c, you can use the formula:
x = -b / (2a)
Let's apply this formula to the given equation: y = -2x^2 + 4x + 1.
Identifying the coefficients:
a = -2
b = 4
c = 1
Now, substitute the values of a and b into the formula:
x = -4 / (2 * -2)
x = -4 / -4
x = 1
The x-coordinate of the vertex is 1. To find the y-coordinate, substitute the value of x back into the equation:
y = -2(1)^2 + 4(1) + 1
y = -2 + 4 + 1
y = 3
Therefore, the vertex is (1, 3). The x-coordinate of the vertex also represents the axis of symmetry. So, the axis of symmetry is x = 1.
In summary, the vertex of the given quadratic equation y = -2x^2 + 4x + 1 is (1, 3), and the axis of symmetry is x = 1.