Problem 66 - Precalculus : Building Concepts and Connections

Question

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\) -intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=-x^{2}+x+3$$

Step-by-Step Solution

Answer

The discriminant of the function \(f(x)=-x^{2}+x+3\) is 13. As it is greater than zero, it shows that the function has two x-intercepts and two real solutions for the equation \(f(x) = 0\).

Step 1: Define the Coefficients

First, identify the coefficients \(a\), \(b\), and \(c\) from the given quadratic function. For the function \(f(x)=-x^{2}+x+3\), the coefficients are:\n\(a=-1\), \(b=1\), and \(c=3\).

Step 2: Calculate the Discriminant

Second, calculate the discriminant of the quadratic function using the formula \(D=b^{2}-4ac\). Insert the coefficients \(a=-1\), \(b=1\), and \(c=3\) into the formula which gives:\n\(D=1^{2}-4*(-1)*3\) = \(1+12=13\).

Step 3: Analyze the Discriminant

The third step is to analyze the result of the discriminant to determine the number of the x-intercepts and the real solutions. If the discriminant is greater than 0, the function has two x-intercepts (or two real solutions of \(f(x)=0\)). If the discriminant equals 0, there is one x-intercept (or one real solution of \(f(x)=0\)). If the discriminant is less than 0, there are no x-intercepts (or no real solutions of \(f(x)=0\)). In our case, the discriminant equals 13, which is greater than 0, so the function has two x-intercepts and two real solutions of the equation \(f(x)=0\).

Key Concepts

Quadratic Functions

Quadratic functions are a key part of algebra and represent a relationship where the output values are based on the squares of the input values. Quadratic functions are generally expressed in the standard form:
\( f(x) = ax^{2} + bx + c \),
where \( a \), \( b \), and \( c \) are constants and \( a \) is not zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient \( a \). When \( a > 0 \), the parabola opens upwards, and when \( a < 0 \), it opens downwards.

In the context of the textbook exercise provided, the quadratic function is given as \( f(x) = -x^{2} + x + 3 \), which means that the parabola of this function will open downwards because the coefficient \( a \) is negative. Understanding the shape and direction of the parabola is crucial in determining the behavior of the function, especially when calculating the function's intercepts and solutions.

X-intercepts of Quadratic Functions

The x-intercepts of a quadratic function, also known as zeros or roots, are the points where the graph crosses the x-axis. These intercepts can be found by solving the equation \( f(x) = 0 \). Since the standard form of a quadratic equation is \( ax^{2} + bx + c = 0 \), finding the x-intercepts requires solving for the variable \( x \).

In our exercise, the coefficients identified were \( a = -1 \), \( b = 1 \), and \( c = 3 \). The discriminant formula, \( D = b^2 - 4ac \), is the first step in finding the x-intercepts. It gives us valuable information about the number and nature of the x-intercepts by evaluating the term under the square root in the quadratic formula, \( x = \frac{-b \pm \sqrt{D}}{2a} \). Depending on the value of the discriminant, we can determine if the x-intercepts are real numbers, and how many there are. If the discriminant is greater than zero, we expect two x-intercepts; if it's zero, there's exactly one x-intercept; and if it's less than zero, the function does not cross the x-axis, indicating there are no real x-intercepts.

Real Solutions of Quadratic Equations

Real solutions of a quadratic equation refer to the values of \( x \) that satisfy the equation \( ax^{2} + bx + c = 0 \) with real numbers. The number of real solutions is directly linked to the discriminant, \( D = b^{2} - 4ac \).

For the given exercise, after computing the discriminant to be 13, which is positive, we can affirm that there are two distinct real solutions. This is directly inferred from the fact that the square root of a positive number is real, resulting in two different solutions when applied to the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \). Given that real solutions also mean the points where the graph intersects the x-axis, the conclusion is that the quadratic function provided will intersect the x-axis at two points. This analysis is crucial for graphing the function accurately and for understanding the function's behavior in different scenarios. Being able to determine the number of real solutions is an essential skill when studying quadratic functions.