Question

What are the key aspects of the graph of f(x) = x2 - 62, where b is a real number?

Answer 1

Answer:

The x intercepts are b , - b

The y-intercept is - b²

The vertex is (0 , - b²)

The axis of symmetry of the function is the y-axis

Step-by-step explanation:

* Lets explain the key aspects of the quadratic function

- There are many key aspects in a quadratic graph such as:

# x-intercepts

# y-intercept

# The vertex

# Axis of symmetry

* Lets solve the problem

∵ f(x) = x² - b² is a quadratic function

∵ The general form of the quadratic function is f(x) = Ax² + B(x) + C

   where A , B , C are constant

∴ A = 1 , B = 0 , C = -b²

- To find the x-intercept put f(x) = 0

∵ f(x) = x² - b²

∴ x² - b² = 0 ⇒ add b² for both sides

∴ x² = b² ⇒ take √ for both sides

∴ x = ± b

∴ The x intercepts are b , - b

- To find the y-intercept put x = 0

∵ f(x) = x² - b²

∴ f(0) = 0 - b²

∴ f(0) = - b²

∴ The y-intercept is - b²

- The vertex of the function is (h , k) where h = -B/2A and k = f(h)

∵ A = 1 and B = 0

∴ h = 0/2(1) = 0

∵ k = f(h)

∴ k = f(0) = 0 - b² = - b²

∴ k = - b²

∴ The vertex is (0 , - b²)

- The axis of symmetry of the quadratic function is a vertical line

  passes through the vertex of it and its equation is x = h

∵ h = 0

∴ The equation of the axis of symmetry is x = 0

∵ The equation of the y-axis is x = 0

∴ The axis of symmetry of the function is the y-axis