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A quadratic equation is any equation in the form y = ax² + bx + c where a, b, and c are constants. Quadratic equations form parabolic shapes when graphed. These equations are crucial for modeling various real-world phenomena like projectile motion and area problems.

In the given equation y = x² - 4x - 12, the coefficients are:

• a = 1
• b = -4
• c = -12

Recognizing these coefficients helps in further calculations like finding intercepts and the vertex, essential steps for graphing the parabola.

This form is often called the 'Standard Form' of a quadratic equation. Understanding the meanings and roles of a, b, and c is foundational to mastering quadratic functions.

Intercepts are points where the graph crosses the axes:

• Y-Intercept: This is found when x is set to 0. Solve y = x² - 4x - 12 at x = 0 to get:
• y = (0)² - 4(0) - 12 = -12
• Thus, the y-intercept is (0, -12).

• X-Intercepts: These are found when y is set to 0. Set y = 0 in y = x² - 4x - 12 and solve the resulting quadratic equation:
• 0 = x² - 4x - 12
• Using the quadratic formula: x = \(\frac{-(-4) ± \sqrt{(-4)² - 4(1)(-12)}}{2(1)}\)
• This resolves to: x = 6 and x = -2
• Thus, the x-intercepts are (6, 0) and (-2, 0).

The vertex of a parabola gives a peak or a trough, representing the maximum or minimum value. For a quadratic equation y = ax² + bx + c, the vertex formula is:

• x = -\(\frac{b}{2a}\)
• For the given equation y = x² - 4x - 12:
• b = -4 and a = 1, so x = -\(\frac{-4}{2(1)}\) = 2
• Then, calculate y at x = 2:
• y = (2)² - 4(2) - 12 = -16

Thus, the vertex is (2, -16).

A parabola is the U-shaped curve of a quadratic function. These curves open upwards if a > 0 and downwards if a < 0. Parabolas have a functional symmetry about their vertex.

To sketch a parabola:

• Identify intercepts like (0, -12), (6, 0), and (-2, 0).
• Find and mark the vertex, in this example, (2, -16).
• Plot these critical points and draw a smooth curve through them, making sure the curve opens upwards due to a being positive.
• Label all key points for clarity.

This method helps visualize quadratic relationships and understand the curve's behavior.