91Ó°ÊÓ

Finding the zeros of a function is a crucial aspect of understanding quadratic equations. These zeros are also known as the roots or x-intercepts of the function. For the quadratic function given by \( f(x) = x^2 - 4x \), the zeros are the values of \( x \) for which \( f(x) = 0 \).
To find these zeros, we can use the quadratic formula, which is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). By substituting the coefficients \( a = 1 \), \( b = -4 \), and \( c = 0 \), we can calculate:
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 0}}{2 \cdot 1} = \frac{4 \pm 4}{2} \]
This simplifies to \( x = 0 \) and \( x = 4 \). These values indicate where the function touches or crosses the x-axis (i.e., the zeros).

• The zeros represent the solutions to the equation \( x^2 - 4x = 0 \).
• They signify points where the graph intersects the x-axis.
• Having multiple zeros may mean the function touches the axis at different points.

Knowing the zeros is helpful when sketching the graph of a quadratic function as it gives clear points of intersection with the horizontal axis.

In a quadratic function, the vertex of a parabola is the peak or the lowest point of the curve. It represents either a maximum or a minimum value of the function, depending on whether the parabola opens upwards or downwards. For the function \( f(x) = x^2 - 4x \), since \( a = 1 \), the parabola opens upwards, indicating that the vertex is a minimum point.

• The vertex \(x\)-coordinate is determined using \( x = -\frac{b}{2a} \), where \( b = -4 \) and \( a = 1 \).
• This calculates to \( x = -\frac{-4}{2 \cdot 1} = 2 \).
• Plug \( x = 2 \) back into the function to find the function value: \( f(2) = 2^2 - 4 \cdot 2 = -4 \).

Thus, the vertex of the parabola is at the point \( (2, -4) \). Understanding the vertex allows us to find the **extreme value** that the function can attain, which is very useful in optimization problems and sketching the graph accurately.

The quadratic formula is a powerful solution tool for finding the roots of any quadratic equation, given in the standard form \( ax^2 + bx + c = 0 \). This elegant formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), provides the x-values where the function intersects the x-axis, solving the equation directly.

• The expression under the square root, \( b^2 - 4ac \), is known as the discriminant. It determines the nature of the solutions:
  • If it is positive, the function has two distinct real roots.
  • If zero, the function has one real repeated root.
  • If negative, the roots are complex (non-real).
• In the exercise \( f(x) = x^2 - 4x \), substituting the coefficients \( a = 1 \), \( b = -4 \), and \( c = 0 \) into the formula provided real-valued roots: \( x = 0 \) and \( x = 4 \).

Mastering the use of the quadratic formula is essential for solving many problems involving quadratic equations, ensuring you can consistently find the zeros of the function efficiently.