Graphing Quadratic Functions
Find any x-intercepts and the y-intercept. If no x-intercepts exist, state this
g(x)=2x^2+3x-1
To find the x-intercepts (also called zeros or roots) and the y-intercept of a quadratic function, you need to follow these steps:
Step 1: Set the function equal to zero, as x-intercepts occur where the function intersects the x-axis:
2x^2 + 3x - 1 = 0
Step 2: Use factoring, completing the square, or the quadratic formula to solve for x. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For g(x) = 2x^2 + 3x - 1, a = 2, b = 3, and c = -1. Plugging these values into the quadratic formula:
x = (-3 ± √(3^2 - 4 * 2 * -1)) / (2 * 2)
x = (-3 ± √(9 + 8)) / 4
x = (-3 ± √17) / 4
This gives us two potential x-intercepts: x = (-3 + √17) / 4 and x = (-3 - √17) / 4.
Step 3: To find the y-intercept, substitute x = 0 into the function:
g(0) = 2(0)^2 + 3(0) - 1
g(0) = -1
Therefore, the y-intercept is -1.
In conclusion, the x-intercepts for g(x) = 2x^2 + 3x - 1 are (-3 + √17) / 4 and (-3 - √17) / 4, and the y-intercept is -1.