For y= 2x squared + x - 1 the x-intercepts and the turning point respectively are?
y = 2x^2 + x - 1
= (2x-1)(x+1)
that shows the x-intercepts
y = 2(x^2 + x/2) - 1
now add and subtract 2/16
y = 2(x^2 + x/2 + 1/16) -1 - 2/16
y = 2(x+1/4)^2 - 9/8
(y+9/8) = 2(x+1/4)^2
that shows where the vertex is. Note that x= -1/4 is halfway between the roots of -1 and 1/2
To find the x-intercepts and the turning point of the equation y = 2x^2 + x - 1, we need to solve a few mathematical equations.
1. X-intercepts:
The x-intercepts are the points where the graph crosses the x-axis, meaning the value of y is zero. To find the x-intercepts, we set y = 0 and solve for x.
0 = 2x^2 + x - 1
To solve this quadratic equation, we can either factorize or use the quadratic formula. In this case, factoring may not be easy, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 2, b = 1, and c = -1.
Substituting these values into the formula, we get:
x = (-(1) ± √((1)^2 - 4(2)(-1))) / (2(2))
Simplifying further:
x = (-1 ± √(1 + 8)) / 4
x = (-1 ± √9) / 4
x = (-1 ± 3) / 4
This gives us two possible x-intercepts:
x1 = (-1 + 3) / 4 = 2 / 4 = 1/2
x2 = (-1 - 3) / 4 = -4 / 4 = -1
Therefore, the x-intercepts are x = 1/2 and x = -1.
2. Turning point:
The turning point of a quadratic function is the vertex of its graph. The x-coordinate of the turning point can be found using the formula:
x = -b / (2a)
Here, a = 2 and b = 1.
x = -(1) / (2(2))
x = -1/4
To find the corresponding y-coordinate, substitute this x value back into the equation:
y = 2x^2 + x - 1
y = 2(-1/4)^2 + (-1/4) - 1
y = 2(1/16) - 1/4 - 1
y = 1/8 - 1/4 - 1
y = 1/8 - 2/8 - 8/8
y = -9/8
Therefore, the turning point is at (-1/4, -9/8).