The solutions to a quadratic equation are -1 and 7. What is the equation of its axis of symmetry?
A little technicality :
A quadratic equation does not have an axis of symmetry, a quadratic function does.
the corresponding quadratic function would be
f(x) = (x+1)(x-7), which would have zeros of -1 and 7
the axis of symmetry would be half-way between these two values, or
x = 4
please look over is it suppose to be 3 if i do -1+7 =6 then divied it by 2 i get 3
You are right, 4 units over from -1 is 3
sorry about the silly arithmetic error.
midpoint between (-1,) and (7,0) is (3,0)
x = 3
The axis of symmetry of a quadratic equation is a vertical line that passes through the vertex of the parabola defined by the equation. To find the equation of the axis of symmetry, we need to find the x-coordinate of the vertex.
Given that the solutions to the quadratic equation are -1 and 7, we know that these are the x-values where the equation equals zero. A quadratic equation in general form is written as ax^2 + bx + c = 0, where a, b, and c are constants.
Since -1 and 7 are the solutions, we can set up two equations:
(-1)^2 + b(-1) + c = 0 -- Equation (1)
(7)^2 + b(7) + c = 0 -- Equation (2)
Simplifying these equations, we get:
1 - b + c = 0 -- Equation (1)
49 + 7b + c = 0 -- Equation (2)
We can solve this system of equations to find the values of b and c by subtracting Equation (1) from Equation (2):
49 + 7b + c - (1 - b + c) = 0
Simplifying, we get:
48 + 8b = 0
Divide both sides by 8:
6 + b = 0
Subtract 6 from both sides:
b = -6
Now, we can substitute the value of b into Equation (1) to find c:
1 - (-6) + c = 0
1 + 6 + c = 0
7 + c = 0
c = -7
So, we have found that b = -6 and c = -7.
Now, we can write the equation of the quadratic function in vertex form, which is given by:
f(x) = a(x - h)^2 + k
Where (h, k) represents the coordinates of the vertex. Since we do not have the direct formula for a quadratic function, we can find the a value using one of the known solutions.
Plugging (-1, 0) into the vertex form, we get:
0 = a(-1 - h)^2 + k
Since (h, k) represents the vertex, we can deduce that -h = -1 and h = 1.
Substituting h = 1 in Equation, we have:
0 = a(-1 - 1)^2 + k
0 = 4a + k
We also know the value of k = -7. Substituting this in, we get:
0 = 4a - 7
Rearranging the equation:
4a = 7
Dividing both sides by 4:
a = 7/4
Now we have the values for a, b, and c. Plugging these back into the general form, we get:
f(x) = (7/4)x^2 - 6x - 7
To find the equation of the axis of symmetry, we use the x-coordinate of the vertex, which is h. From earlier, we found that h = 1. Therefore, the equation of the axis of symmetry is x = 1.