Consider the following claim: if the point (2 + d, y) is on the graph of the function
f(x) = x(x-4), then the point (2 - d, y) is also on the graph.

use algebra to show that the claim is true
what is the relationship between the line x = 2 and the graph of f(x)? justify your reasoning.
show steps

The point (2 - d, y) is on the graph of f(x)

The line x = 2 is the axis of symmetry of the graph of f(x)

Step-by-step explanation:

* Lets explain how to prove that a point lies on a graph Algebraically

- Substitute the value of the x-coordinate of the point in the equation

 of the graph the answer must be equal the y-coordinate of the point

- The function is a quadratic because the greatest power of x is 2,

  then it represented by parabola

- The parabola has a vertex point (h , k), where h is the x-coordinate

  and k is the y-coordinate

- This vertex divides the parabola into two equal parts, then the axis

 of symmetry of the parabola is a vertical line passing through it

∴ The equation of the axis of symmetry is x = h

- The vertex of the parabola could be minimum point if the parabola

  opened upward or maximum if it opened downward

- The minimum value and the maximum value are the value of k

# Look to the attached figures for more understand

* Now lets solve the problem

∵ f(x) = x(x - 4)

∵ Point (2 + d , y) is on the graph of f(x)

- Replace each x in f(x) by 2 + d

∴ f(2 + d) = (2 + d)(2 + d - 4) ⇒ add 2 and -4

∴ f(2 + d) = (2 + d)(-2 + d)

∵ f(2 + d) = y

∴ y = (2 + d)( -2 + d)

* Multiply them to simplify

∴ y = 2(-2) + 2(d) + d(-2) + d(d) = -4 + 2d - 2d + d²

∴ y = -4 + d²

* Lets do these steps again with point (2 - d , y)

- Replace each x in f(x) by 2 - d

∴ f(2 - d) = (2 - d)(2 - d - 4) ⇒ add 2 and -4

∴ f(2 - d) = (2 - d)(-2 - d)

∵ f(2 - d) = y

∴ y = (2 - d)( -2 - d)

* Multiply them to simplify

∴ y = 2(-2) + 2(-d) - d(-2) - d(-d) = -4 - 2d + 2d + d²

∴ y = -4 + d²

- The value of y of the point (2 - d , y) = the value of y of the point on

  the graph

∵ f(2 + d) = f(2 - d)

∵ The point (2 + d , y) is on the graph of f(x)

∴ The point (2 - d , y) is on the graph of f(x)

* It is true the point (2 - d, y) is also on the graph.

* To find the relation between the line x = 2 and the graph of f(x)

  lets find the vertex of the parabola

- If f(x) = ax² + bx + c in the general form, where a, b , c are constant

 then h = -b/2a, where h is the x-coordinate of the vertex point, a is

 the coefficient of x² and b is the coefficient of x

∵ f(x) = x(x - 4) ⇒ multiply the bracket by x to put it in the general form

∴ f(x) = x² - 4x

- Find the value of a and b to find h

∵ a = 1 and b = -4

∵ h = -b/2a

∴ h = -(-4)/2(1) = 4/2 = 2

∴ The x-coordinate of the vertex point = 2

∵ The axis of symmetry of the parabola passing through the

   vertex point

∴ The equation of the axis of symmetry of the parabola is x = 2

* The line x = 2 is the axis of symmetry of the graph of f(x)

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step-by-step explanation:

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step-by-step explanation:

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