I am totally lost on how to solve
g(x)=2x^2+3x-1 to find the y and x-intercepts.
This is what I have so far, but I don't know if I am on the right track. And if so what would be next?
g(x)=2x^2+3x-1
=2x^2+3x -1
=(2x^2+3+6) +3-1
Get y-intercept when x=0
g(x) = y = 2(0)^2 + 3(0) - 1
(0,-1) ; y-intercept = -1
x-intercept: set function equal to 0
2x^2 + 3x - 1 = 0
After quadratic equation...
x-intercepts: x = -1.78, and x = 0.28
function to the 2nd power of x, so is a parabola opening upwards. Graphing the function verifies the answers.
To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x.
Let's start by setting g(x) equal to zero:
2x^2 + 3x - 1 = 0
To solve this equation, you have a few options. One common way is to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 3, and c = -1. Plugging these values into the quadratic formula, you get:
x = (-3 ± √(3^2 - 4(2)(-1))) / (2(2))
x = (-3 ± √(9 + 8)) / 4
x = (-3 ± √17) / 4
So the x-intercepts are (-3 + √17)/4 and (-3 - √17)/4.
To find the y-intercept, you can substitute x = 0 into the original equation:
g(0) = 2(0)^2 + 3(0) - 1
g(0) = 0 + 0 - 1
g(0) = -1
Therefore, the y-intercept is -1.
In summary:
- x-intercepts: (-3 + √17)/4 and (-3 - √17)/4
- y-intercept: -1
To find the x-intercepts, you need to set the function equal to zero and solve for x. In other words, you are looking for the values of x where g(x) intersects or crosses the x-axis.
So, to find the x-intercepts of the function g(x) = 2x^2 + 3x - 1, you need to set g(x) equal to zero:
2x^2 + 3x - 1 = 0
Now, you have a quadratic equation. To solve it, you can use several methods, such as factoring, completing the square, or using the quadratic formula.
In this case, factoring may not be straightforward, so let's solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In the equation, a, b, and c come from the standard form of a quadratic equation, which is ax^2 + bx + c = 0.
In this case, a = 2, b = 3, and c = -1:
x = (-(3) ± √((3)^2 - 4(2)(-1))) / (2(2))
Now, simplify the equation:
x = (-3 ± √(9 + 8)) / 4
x = (-3 ± √(17)) / 4
These are the solutions for the x-values where g(x) intersects the x-axis, also known as the x-intercepts. You can leave the answer in this exact form, or you can approximate it as a decimal if necessary.
To find the corresponding y-intercepts, you need to substitute the x-values you just found back into the original equation g(x). The y-intercept is the point on the y-axis where the graph of a function intersects.
To find the y-intercept, substitute x = 0 into the equation g(x):
g(0) = 2(0)^2 + 3(0) - 1
g(0) = 0 + 0 - 1
g(0) = -1
Therefore, the y-intercept is -1. The coordinates of the y-intercept are (0, -1), where x = 0 and y = -1.