Problem 18 - Introductory Algebra for College Students

Question

Find the \(y\) -intercept for the parabola whose equation is given. $$y=x^{2}+8 x$$

Step-by-Step Solution

Answer

The y-intercept of the parabola is \(y = 0\).

Step 1: Identify the Equation

The given equation is \(y = x^{2} + 8x\). The y-intercept will be found by setting \(x = 0\) in the equation.

Step 2: Substitute x = 0 into the equation

Substitute \(x = 0\) into the given equation. This gives us: \(y = (0)^{2} + 8*0\).

Step 3: Solve for y

Upon evaluating the expression, we find that \(y = 0\). Therefore, the y-intercept of the parabola is \(y = 0\)

Key Concepts

Parabola Equation

In algebra, a parabola is a curve that can be expressed by a quadratic equation. A standard parabola equation is usually written in the form of \(y = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants that determine the shape and position of the parabola. In our example given as \(y = x^2 + 8x\), we notice that it fits this structure with \(a = 1\), \(b = 8\), and \(c = 0\).
The absence of \(c\) means that the parabola's vertex does not have an additional vertical shift along the y-axis. When dealing with a parabola, the coefficients \(a\) and \(b\) control aspects such as the direction of the opening and the location of the axis of symmetry.
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.

Algebra

Algebra is the branch of mathematics in which numbers and quantities are represented by letters and symbols. This allows us to form equations that can describe patterns and solve problems. In the context of the parabola equation provided, algebraic manipulation is used to find specific points, like intercepts, that help us understand the graph's features.
It's essential to understand the basic operations and principles in algebra to tackle such equations effectively.

• Equations involve expressions set to be equal to each other, often solved by performing operations that simplify or modify these expressions.
• Algebra utilizes both known and unknown values, with the goal of finding unknowns (like variables within equations).

Mastering how these elements interact can be immensely beneficial in solving quadratic equations like the one in our exercise.

Finding Intercepts

Intercepts are points where a graph crosses the axes. For parabolas, these include the y-intercept and potentially x-intercepts (or roots). Discovering intercepts adds context and clarity to understanding a parabola's behavior on a graph.

- **Y-intercept:** This is the value of \(y\) when \(x = 0\). By setting \(x = 0\) in the equation, you solve for \(y\). In this specific exercise, substituting \(x = 0\) into \(y = x^2 + 8x\) results in \(y = 0\), identifying the y-intercept as \( (0, 0) \).
- **X-intercepts (Roots):** Set \(y = 0\) to find where the graph intersects the x-axis. Solving \(x^2 + 8x = 0\) provides any x-intercepts.

These intercepts are useful for sketching graphs and recognizing where a parabola lies in reference to the coordinate plane.