PLEASE HELP A sketch of the graph y = 2x² - 4x – 7 is shown.
Find the values identified as a, b, c, d and e on the sketch.

a = (0, -7)

b = (1, 0)

c = (-9, 0)

d = (1 - 3√2 / 2, 0)  ≈ (-1.1213, 0)

e = (1 + 3√2 / 2, 0) ≈ (3.1213, 0)

Step-by-step explanation:

points d and e

d and e we can find by solving for y = 0:

2x² - 4x - 7 = 0

2x² - 4x = 7

x² - 2x = 7/2

x² - 2x + 1 = 9/2

(x - 1)² = 9/2

x - 1 = √(9/2)

x = 1 ± 3√2 / 2

x ≈ 1 ± 2.1213

x ≈ 3.1213, -1.1213

So d is at approximately (-1.1213, 0), and e (3.1213, 0)

--- point a ---

Much easier to do, we simply need to solve for x = 0:

y = 2 × 0² - 4 × 0 - 7

y = -7

So point a is at (0, -7)

--- points b and c ---

We can find these in a couple of interesting ways. We've actually found b already, as it's halfway between points d and e.  You'll recall that those are calculated as 1 ± 3√2 / 2, meaning that the midway point between them is just 1.

An alternate way of doing it is to apply some basic calculus, finding the point at which the curve has a slope of zero, which will give us b's x coordinate:

y = 2x² - 4x - 7

dy/dx = 4x - 4

0 = 4x - 4

4x = 4

x = 1

So b is equal to (1, 0).

To find c, we simply plug that x = 1 into the original function:

y = 2x² - 4x - 7

y = 2(1)² - 4(1) - 7

y = 2 - 4 - 7

y = -9

so c lies on (0, -9)