Given the quadratic function y =3x^2+10x-8 (i)Find the y-intercept. (ii)Find the x-intercept. (iii) Calculate the axis of symmetry. (iv)Find the coordinates of the vertex (v)Sketch the graph of the above function.

To find the y-intercept, we need to determine the value of y when x is 0.

Substitute x = 0 into the quadratic function:
y = 3(0)^2 + 10(0) - 8
y = 0 + 0 - 8
y = -8
Therefore, the y-intercept is (0, -8).

To find the x-intercepts, we need to determine the values of x where the function intersects the x-axis, meaning y = 0.
Set the quadratic function to zero:
0 = 3x^2 + 10x - 8
We can solve this quadratic equation in several ways, such as factoring, completing the square, or using the quadratic formula. In this case, we will use factoring.
First, rearrange the equation:
3x^2 + 10x - 8 = 0
Next, try to factor the quadratic equation:
(3x - 2)(x + 4) = 0
By setting each factor to zero, we get two possible solutions:
3x - 2 = 0 or x + 4 = 0
3x = 2 or x = -4
x = 2/3 or x = -4
Therefore, there are two x-intercepts: (2/3, 0) and (-4, 0).

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It can be found using the formula: x = -b/ (2a), where a, b, and c are coefficients in the quadratic equation.
For the given quadratic function y = 3x^2 + 10x - 8, a = 3 and b = 10.
Using the formula, we can calculate the axis of symmetry as follows:
x = -(10) / (2 * 3)
x = -10 / 6
x = -5 / 3
Therefore, the axis of symmetry is x = -5/3.

To find the coordinates of the vertex, we can use the axis of symmetry to determine the x-coordinate and substitute it into the quadratic function to find the y-coordinate.
From the previous step, we found that the x-coordinate of the vertex is -5/3.
Substituting x = -5/3 into the quadratic function:
y = 3(-5/3)^2 + 10(-5/3) - 8
Simplifying further:
y = 3(25/9) - 50/3 - 8
y = 75/9 - 50/3 - 8
y = 25/3 - 50/3 - 8
y = -14/3
Therefore, the coordinates of the vertex are (-5/3, -14/3).

To sketch the graph of the quadratic function y = 3x^2 + 10x - 8, we have the following information:
- The y-intercept is (0, -8).
- The x-intercepts are (2/3, 0) and (-4, 0).
- The axis of symmetry is x = -5/3.
- The vertex is (-5/3, -14/3).

Plot these points on a coordinate plane and draw a smooth curve passing through them. The graph of the quadratic function y = 3x^2 + 10x - 8 should be a U-shaped curve opening upwards.