Determine the vertex, axis of symmetry, x-intercept, and y -intercept for the equation: y=−2x^2 + 12x − 13 .
To determine the vertex, axis of symmetry, x-intercept, and y-intercept for the equation y = −2x^2 + 12x − 13, we can start by finding the vertex.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the x^2 and x terms respectively.
From the equation y = −2x^2 + 12x − 13, we can see that a = -2 and b = 12. Plugging these values into the formula, we get x = -12 / (2 * -2) = 12 / 4 = 3.
To find the y-coordinate of the vertex, we substitute the x-coordinate back into the equation:
y = −2(3)^2 + 12(3) − 13 = -18 + 36 - 13 = 5.
Therefore, the vertex is (3, 5).
The axis of symmetry is the vertical line that passes through the vertex. Since the x-coordinate of the vertex is 3, the equation of the axis of symmetry is x = 3.
To find the x-intercepts, we set y = 0 and solve the equation:
0 = −2x^2 + 12x − 13.
This is a quadratic equation. We can use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the solutions.
For our equation, a = -2, b = 12, and c = -13. Plugging these values into the formula, we get:
x = (-12 ± √(12^2 - 4 * -2 * -13)) / (2 * -2)
= (-12 ± √(144 - 104)) / (-4)
= (-12 ± √(40)) / (-4)
= (-12 ± 2√10) / (-4)
= 3 ± (√10 / 2).
The x-intercepts are therefore x = 3 + (√10 / 2) and x = 3 - (√10 / 2).
To find the y-intercept, we substitute x = 0 into the equation:
y = −2(0)^2 + 12(0) − 13 = -13.
Therefore, the y-intercept is (0, -13).
In summary, the vertex is (3, 5), the axis of symmetry is x = 3, the x-intercepts are approximately 3 + (√10 / 2) and 3 - (√10 / 2), and the y-intercept is (0, -13).