Problem 17 - Precalculus

Question

Determine any \(x\) - or \(y\) -intercepts for the graph of the equation. Note: You're not asked to draw the graph. (a) \(y=x^{2}+3 x+2\) (b) \(y=x^{2}+2 x+3\)

Step-by-Step Solution

Answer

(a) X-intercepts: (-1, 0), (-2, 0); Y-intercept: (0, 2). (b) No real x-intercepts; Y-intercept: (0, 3).

Step 1: Identify the Intercepts for (a)

For part (a), the equation is given as \( y = x^2 + 3x + 2 \). We need to find both x-intercepts and the y-intercept. Start with the y-intercept by setting \( x = 0 \).

Step 2: Calculate the y-intercept for (a)

Substitute \( x = 0 \) into the equation: \( y = 0^2 + 3 \cdot 0 + 2 = 2 \). Therefore, the y-intercept is \( (0, 2) \).

Step 3: Calculate the x-intercepts for (a)

Set \( y = 0 \) to find the x-intercepts: \( 0 = x^2 + 3x + 2 \). Factor the quadratic equation: \( (x+1)(x+2) = 0 \). Solve each factor: \( x+1=0 \) gives \( x = -1 \), and \( x+2=0 \) gives \( x = -2 \). So, the x-intercepts are \( (-1, 0) \) and \( (-2, 0) \).

Step 4: Identify the Intercepts for (b)

For part (b), the equation is \( y = x^2 + 2x + 3 \). Again, we find both x-intercepts and the y-intercept. Start with the y-intercept by setting \( x = 0 \).

Step 5: Calculate the y-intercept for (b)

Substitute \( x = 0 \) into the equation: \( y = 0^2 + 2 \cdot 0 + 3 = 3 \). Therefore, the y-intercept is \( (0, 3) \).

Step 6: Attempt to Calculate x-intercepts for (b)

Set \( y = 0 \) to find the x-intercepts: \( 0 = x^2 + 2x + 3 \). Attempt to factor the quadratic, but recognize it can't be factored easily. Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, \) and \( c = 3 \).

Step 7: Apply the Quadratic Formula for (b)

Calculate the discriminant: \( b^2 - 4ac = 2^2 - 4(1)(3) = 4 - 12 = -8 \). Since the discriminant is negative, there are no real x-intercepts.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable, usually denoted as \( x \), is 2. They are typically written in the standard form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).

Understanding the properties of quadratic equations is crucial for solving problems related to graphs, as their shape and position depend on the values of \( a \), \( b \), and \( c \).

Parabolas have very characteristic points called intercepts. The **y-intercept** is where the graph crosses the y-axis and is found by setting \( x = 0 \). The **x-intercepts** are points where the graph crosses the x-axis, found by setting \( y = 0 \) and solving the quadratic equation.

Factoring Quadratics

Factoring is one of the methods used to solve quadratic equations, particularly when searching for x-intercepts. It's a technique that transforms the quadratic expression into a product of two or more simple expressions. To factor a quadratic like \( y = x^2 + 3x + 2 \), you need to find two numbers that multiply to give the constant term, \( c \), and add to give the linear coefficient, \( b \).

For example, in the quadratic \( y = x^2 + 3x + 2 \), we find that the numbers 1 and 2 multiply to 2 and add to 3, allowing us to factor the expression as \( (x+1)(x+2) = 0 \).

Factoring is quite efficient for quadratics that can be broken down easily. However, when a quadratic equation does not factor nicely, other methods like completing the square or using the quadratic formula become necessary.

Quadratic Formula

The quadratic formula is a universal solution for any quadratic equation, especially those that cannot be factored easily. It is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula provides the solutions for \( x \) by substituting the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \).

Using the quadratic formula can determine not just the real roots or x-intercepts, but also indicate the nature of these roots based on the discriminant. For instance, in the equation \( y = x^2 + 2x + 3 \), substituting \( a = 1 \), \( b = 2 \), and \( c = 3 \) into the formula helps determine the behavior of the parabola, even when real x-intercepts do not exist.

Discriminant in Quadratics

The discriminant is a key part of the quadratic formula, calculated as \( b^2 - 4ac \). It provides crucial information about the number and type of solutions, or roots, that a quadratic equation will have.

The value of the discriminant tells us:

• If it is positive, the quadratic equation has two distinct real roots (x-intercepts). This means the parabola will intersect the x-axis at two points.
• If it is zero, there is exactly one real root, meaning the parabola touches the x-axis at a single point, called a vertex.
• If it is negative, as seen in the problem \( y = x^2 + 2x + 3 \), there are no real roots. The parabola will not intersect the x-axis, indicating the presence of complex roots instead.

Understanding the discriminant helps in predicting graph behaviors without plotting, allowing one to determine the existence of intercepts swiftly.