apply the quadratic formula to find the roots of the given function, and then graph the function.
f(x) = x2 - 4
g(x) = x2 - x - 12
see response on other post
To apply the quadratic formula to find the roots of a given function, we need to have the function in the form of ax^2 + bx + c = 0.
Let's start by applying the quadratic formula to the function f(x) = x^2 - 4:
1. Identify the values of a, b, and c. In this case, a = 1, b = 0, and c = -4.
2. Plug these values into the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
For f(x), the equation becomes x = (-0 ± √(0^2 - 4(1)(-4))) / (2(1)).
Simplifying this equation gives us x = ±2.
Therefore, the roots or x-intercepts of the function f(x) = x^2 - 4 are x = 2 and x = -2.
Now let's move on to the function g(x) = x^2 - x - 12:
1. Again, identify the values of a, b, and c. In this case, a = 1, b = -1, and c = -12.
2. Apply the quadratic formula: x = (-(-1) ± √((-1)^2 - 4(1)(-12))) / (2(1)).
Simplifying this equation gives us x = (1 ± √(1 + 48)) / 2.
Further simplification gives us x = (1 ± √49) / 2, which gives us two possible solutions:
x = (1 + 7) / 2 = 4 and x = (1 - 7) / 2 = -3.
Therefore, the roots or x-intercepts of the function g(x) = x^2 - x - 12 are x = 4 and x = -3.
To graph the given functions, f(x) and g(x), we can plot the points using the roots or x-intercepts we just found and then sketch the curves. We can also find the vertex, y-intercept, and other features by analyzing the equations.
For f(x), the vertex is (0, -4) since the equation is of the form x^2 + c, where c represents the y-coordinate of the vertex.
The y-intercept is (0, -4). Using this information, we can plot these points on a graph and sketch the curve accordingly.
For g(x), we can find the vertex by using the formula x = -b / (2a), which gives us x = -(-1) / (2*1) = 1/2. Plugging this value into the equation gives us g(1/2) = 1/4 - 1/2 - 12 = -48/4 = -12. So the vertex is (1/2, -12).
The y-intercept is (0, -12). Using this information, we can plot these points on the graph and sketch the curve accordingly.
Note: It's always helpful to plot additional points to get a better understanding of the shape of the graph.
I hope this helps you understand how to apply the quadratic formula to find the roots and graph quadratic functions!