Given the quadratic equation y = -x^2 + 2x + 3.

a) Will the graph turn upwards or downwards?
b) What are the coordinates of x and y intercepts?
c) What is the coordinate of the turning point?
d) Use your answer in parts (a,b,c and d) to sketch the graph.

(a) the -x^2 means it opens downward

(b) set x=0 to get y-intercept
Since y = (1-x)(x-3) the x-intercepts are easy to see.
(c) as usual, the vertex is at x = -b/2a = 1. Find y(1).

To answer each part of the question:

a) To determine whether the graph of the quadratic equation y = -x^2 + 2x + 3 turns upwards or downwards, we need to look at the coefficient of the x^2 term. In this case, the coefficient is -1, which is negative. A negative coefficient of the x^2 term indicates that the graph will open downwards.

b) To find the x-intercepts, we need to set y equal to zero and solve for x. So, we set -x^2 + 2x + 3 = 0 and solve the quadratic equation. We can factor it or use the quadratic formula to find the solutions. In this case, factoring would be a bit challenging, so we can use the quadratic formula. The quadratic formula states that x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a, b, c are the coefficients of the quadratic equation. In our equation, a = -1, b = 2, and c = 3. Plugging these values into the formula, we get:

x = (-2 ± sqrt(2^2 - 4(-1)(3))) / (2(-1))
x = (-2 ± sqrt(4 + 12)) / -2
x = (-2 ± sqrt(16)) / -2
x = (-2 ± 4) / -2
x = (-2 + 4) / -2 or x = (-2 - 4) / -2
x = 2 / -2 or x = 6 / -2
x = -1 or x = -3

So, the x-intercepts are -1 and -3. To find the y-intercept, we can set x = 0 in the equation y = -x^2 + 2x + 3 and solve for y:

y = -(0)^2 + 2(0) + 3
y = 3

So, the y-intercept is 3.

c) The x-coordinate of the turning point (also known as the vertex) can be found using the formula x = -b / (2a). In our equation, a = -1 and b = 2, so the x-coordinate of the turning point is x = -(2) / (2*(-1)) = 1.

To find the y-coordinate of the turning point, substitute the x-coordinate (1) back into the equation y = -x^2 + 2x + 3:

y = -(1)^2 + 2(1) + 3
y = -1 + 2 + 3
y = 4

So, the coordinate of the turning point is (1, 4).

d) To sketch the graph, we have the following information:
- The graph opens downwards.
- The x-intercepts are -1 and -3.
- The y-intercept is 3.
- The turning point is at (1, 4).

You can plot these points on a Cartesian plane and sketch the parabolic curve. Start from the y-intercept (0, 3), then plot the x-intercepts (-1, 0) and (-3, 0). Finally, mark the turning point (1, 4). Connect the points smoothly to trace the curve. Remember that the curve opens downwards.

The final sketch of the graph should look like a parabolic curve opening downwards with the x-intercepts at -1 and -3, the y-intercept at 3, and the turning point at (1, 4).