Problem 108 - Algebra and Trigonometry

Question

Crosses the \(x\)-axis at \(-4,0,\) and \(3 ;\) lies above the \(x\)-axis between \(-4\) and \(0 ;\) lies below the \(x\)-axis between 0 and 3

Step-by-Step Solution

Answer

The equation that satisfies the given conditions is of the form \(f(x) = a(x+4)(x-3)\) for any \(a<0\)

Step 1: Formulate the function

Since the roots of the function are given as -4 and 3, a quadratic equation f(x) can be formulated as:\[f(x) = a \cdot (x+4)(x-3)\] where \(a\) is any non-zero real number. The exact value of coefficient \(a\) determines the direction of the opening of the parabola.

Step 2: Determine the direction of the function

The parabola opens upwards if \(a>0\) and downwards if \(a<0\). The problem mentions that the function lies above the x-axis between -4 and 0. If we test with a value between -4 and 0, say -2, into the equation, the sign of \(f(-2)\) should be positive. Plugging into the equation we get:\[f(-2) = a(-2 + 4)(-2 - 3) = a \cdot 2 \cdot -5 = -10a\] Since \(f(-2)\) needs to be positive and -10a is negative, a needs to be negative to make the equation positive. Therefore, the value of \(a\) must be less than zero, meaning the function opens downwards.

Step 3: Final Function

With everything taken into account, our quadratic function should be:\[f(x) = a(x+4)(x-3)\] for any \(a<0\), which satisfies the initial given conditions.

Key Concepts

Roots of a Quadratic Equation

The roots of a quadratic equation, also known as solutions or x-intercepts, are the values of 'x' for which the quadratic function equals zero. For a quadratic equation in standard form \( ax^2 + bx + c = 0 \), finding the roots involves solving for 'x'.

• These can be found using the factoring method, completing the square, the quadratic formula, or graphically if the equation represents a parabola.
• When given the roots, like -4 and 3 in our exercise, we can construct the quadratic function as \( f(x) = a(x+4)(x-3) \) because a quadratic equation with roots \( r_1 \) and \( r_2 \) is given by \( f(x) = a(x - r_1)(x - r_2) \).
• In the exercise, the coefficient 'a' must be negative to ensure that the parabola opens downwards, as indicated by the function lying below the x-axis between 0 and 3.

Recognizing the role of roots in building a quadratic equation not only helps in solving for 'x', but also forms the foundation for understanding the graph of the function represented by the equation.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function. Every parabola has unique features characterized by its orientation, width, and the position of its vertex (the highest or lowest point on the graph, depending on the orientation).

• Its orientation (opening upwards or downwards) is determined by the sign of the coefficient 'a' in the function \( f(x) = ax^2 + bx + c \).
• If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
• In our exercise example, we inferred that since the parabola lies below the x-axis between 0 and 3, it opens downwards, so 'a' is negative. This conclusion helps to visually understand how the quadratic function behaves and guides us in graphing it accurately.

Understanding the characteristics of a parabola is essential in comprehending not just their graphical representation, but also how different pieces of information about the graph relate to the algebraic form of the quadratic equation.

X-Intercepts

The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. These points correspond to the roots of the quadratic equation.

• To find the x-intercepts algebraically, you set the quadratic function equal to zero and solve for 'x'.
• In the context of our exercise, after determining that the function crosses the x-axis at \( x = -4 \) and \( x = 3 \) and opens downwards, we can construct the function based on these intercepts and then further refine the value of 'a' as required.
• Graphically, the x-intercepts are where the parabola touches the x-axis, which in this case is at two points. The knowledge of these points helps us in plotting the curve correctly.

The significance of x-intercepts in the study of quadratic functions extends beyond just finding the roots; they offer insight into the function's graph and can serve as a checkpoint to determine if we have the correct graph corresponding to a given quadratic equation.