Sketch the graph of this quadratic function: f(x)= 3(x-2)^2 -10. Be sure to explain the steps you take.
**I don't need help with the graph part but I need to find the x and y intercepts, the equation of the axis of symmetry and the coordinates of the vertex from the function above. If someone could please provide help, that would be great. Thanks in advance.
f(x) = 3x^2 - 12x + 2
the y-intercept is clearly at 2.
x-intercepts? where y=0. Use the quadratic formula to get
x = 2±√(10/3)
To find the x-intercepts of the quadratic function f(x) = 3(x-2)^2 - 10, we need to set f(x) equal to zero and solve for x. The x-intercepts occur at the points where the graph intersects the x-axis, meaning the y-coordinate is zero.
Set f(x) = 3(x-2)^2 - 10 equal to zero:
3(x-2)^2 - 10 = 0
Next, we can add 10 to both sides to get rid of the -10 term:
3(x-2)^2 = 10
Now, divide both sides by 3 to isolate the (x-2)^2 term:
(x-2)^2 = 10/3
To solve for x, we can take the square root of both sides:
x-2 = ±√(10/3)
Now, we can add 2 to both sides to solve for x:
x = 2 ± √(10/3)
These are the x-intercepts of the quadratic function f(x).
To find the y-intercept, we can simply substitute x = 0 into the function f(x) = 3(x-2)^2 - 10:
f(0) = 3(0-2)^2 - 10
f(0) = 3(-2)^2 - 10
f(0) = 3(4) - 10
f(0) = 12 - 10
f(0) = 2
Therefore, the y-intercept is (0, 2).
The equation of the axis of symmetry can be found using the formula x = -b/2a, where the quadratic function is in the form of f(x) = ax^2 + bx + c. In this case, our function is f(x) = 3(x-2)^2 - 10.
We can see that a = 3 and b = 0, so substituting these values into the formula:
x = -(0) / 2(3)
x = 0 / 6
x = 0
Therefore, the equation of the axis of symmetry is x = 0.
Lastly, to find the coordinates of the vertex, we can use the axis of symmetry and substitute it into the quadratic function. In this case, the x-coordinate of the vertex is 0.
Substitute x = 0 into the function f(x) = 3(x-2)^2 - 10:
f(0) = 3(0-2)^2 - 10
f(0) = 3(-2)^2 - 10
f(0) = 3(4) - 10
f(0) = 12 - 10
f(0) = 2
Therefore, the coordinates of the vertex are (0, 2).