Problem 57 - Algebra 2

Question

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=-2 x^{2}+3 x-1 $$

Step-by-Step Solution

Answer

The function has two distinct real x-intercepts.

Step 1: Determine the Discriminant

To determine the number of x-intercepts of a quadratic function in the form of \(y = ax^2 + bx + c\), we need to calculate the discriminant \(D = b^2 - 4ac\). For the given quadratic function \(y = -2x^2 + 3x - 1\), the coefficients are \(a = -2\), \(b = 3\), and \(c = -1\). Plugging these values into the discriminant formula, we get \(D = (3)^2 - 4(-2)(-1) = 9 - 8 = 1\).

Step 2: Analyze the Discriminant

Now that we have the discriminant, we can determine the number of x-intercepts. If the discriminant is positive, there are two distinct real x-intercepts. If it is zero, there is exactly one real x-intercept (the graph touches the x-axis). If negative, there are no real x-intercepts (the graph does not touch the x-axis). As we calculated that \(D = 1\), which is positive, there are two distinct real x-intercepts for the given quadratic equation.

Key Concepts

Discriminant of a Quadratic Equation

In algebra, the discriminant of a quadratic equation is a powerful tool that helps us understand key properties of the equation's graph without needing to plot it. The discriminant, often symbolized as \( D \), is part of the quadratic formula and is found using the coefficients of the quadratic equation \( ax^2 + bx + c \). The formula for the discriminant is \( D = b^2 - 4ac \).

The value of the discriminant can tell us whether the equation has two distinct real solutions, one real solution, or no real solutions. Here's the breakdown:

• If \( D > 0 \), the equation has two distinct real x-intercepts (the parabola crosses the x-axis at two points).
• If \( D = 0 \), there is exactly one real x-intercept (the parabola touches the x-axis at one point).
• If \( D < 0 \), there are no real x-intercepts (the parabola does not intersect the x-axis).

This means the discriminant gives us a quick way to determine how many times the graph of the quadratic equation will intersect the x-axis.

Quadratic Function

A quadratic function is a type of polynomial function that can be written in the standard form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not equal to zero. The graph of a quadratic function is a curve called a parabola. Depending on the value of \( a \), the parabola opens upwards \( (a > 0) \) or downwards \( (a < 0) \).

The quadratic function has various features, including a vertex (the highest or lowest point of the parabola), an axis of symmetry (a vertical line that divides the parabola into two mirror images), and the previously mentioned x-intercepts (where the graph crosses the x-axis). Using the vertex formula and the discriminant of the quadratic equation, we can find the vertex and predict the number of x-intercepts without graphing the function.

Number of X-Intercepts

The number of x-intercepts a quadratic function has is directly related to the discriminant of the corresponding quadratic equation. As discussed, the discriminant \( D \), calculated by the formula \( D = b^2 - 4ac \), determines the nature and number of x-intercepts.

An important aspect to remember is that the x-intercepts are the points where the parabola crosses the x-axis, and they represent the real solutions of the equation. Knowing the number of x-intercepts can be valuable for various applications, such as solving problems related to projectile motion, maximizing area, or optimizing other practical situations where a maximal or minimal value is sought.

In classroom or homework settings, understanding how to use the discriminant to predict the number of x-intercepts can greatly simplify the process of characterizing the function's graph and can aid students in solving equations without the need for complex graphing.