Answer:
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)
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)
A coordinate system in geometry is a method for determining the precise location of points or other geometrical objects on a manifold, such as Euclidean space, using one or more numbers, or coordinates.
Given
----- 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)
To learn more about co-ordinate geometry refer to:
https://brainly.com/textbook-solutions/q-use-figure-shown-exercises-14-17-example-14
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