91Ó°ÊÓ

A quadratic function is a type of polynomial function where the highest exponent of the variable is 2. The general form of a quadratic function is given by:

f(x) = ax^2 + bx + c

Here, a, b, and c are constants. The term ax^2 is the quadratic term, bx is the linear term, and c is the constant term.

Some important properties of quadratic functions include:

• The graph of a quadratic function is a parabola.
• If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
• The vertex of the parabola is the highest or lowest point on the graph.
• The parabola is symmetric about its vertex.

In the given exercise, we start with the quadratic function:

f(x) = x^2 - 10x - 22

Combined, these regions help explain the shape and position of a quadratic function's graph.

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. At these points, the output value (function value) is zero. For a given function f(x), the x-intercepts can be found by solving the equation:

f(x) = 0

In the context of our exercise, to find the x-intercepts, we set the quadratic function equal to zero and solve for x: f(x) = x^2 - 10x - 22 = 0

To solve this, we completed the square, allowing us to transform the equation into a solvable format. After completing the square and taking square roots, we arrived at the solutions:

x = 5 ± √47

Thus, the x-intercepts of the function f(x) = x^2 - 10x - 22 are given by:

x_1 = 5 + √47
x_2 = 5 - √47
These are the points where the function crosses the x-axis. Plugging these values back into the function would set f(x) to 0.

The square root is a mathematical function that, when applied to a number, produces another number which, when multiplied by itself, gives the original number. The square root of a number p is written as √p. For example, the square root of 9 is 3, because 3 × 3 = 9. Similarly:

• The square root of 25 is 5.
• The square root of 1 is 1.
• The square root of 0 is 0.
• The square root of 47 remains as √47 since 47 is not a perfect square.

In our exercise, we arrived at the equation:

(x - 5)^2 = 47

To solve for x, we take the square root of both sides: x - 5 = ±√47.

Adding 5 to both sides, we get:

x = 5 ± √47

Using square roots helped us solve the quadratic equation. This method is crucial for finding solutions that are not immediately obvious by simple factoring, especially when dealing with imperfect squares.