Problem 49 - Algebra for College Students

Question

Vertex \((-9,1)\), symmetric with respect to the line \(x=-9\), and contains the point \((-8,0)\) $$ x^{2}+18 x+y+80=0 $$

Step-by-Step Solution

Answer

The parabola's vertex is (-9,1) and it includes the point (-8,0), verifying all given conditions.

Step 1: Identify the form of the parabola equation

The given equation of the parabola is \(x^2+18x+y+80=0\). To identify its form, rewrite it as \(y = -x^2 - 18x - 80\), which shows that the parabola opens downwards since the coefficient of \(x^2\) is negative.

Step 2: Verify the vertex

The standard form of a parabola is \(y = ax^2 + bx + c\). The vertex \((h, k)\) of a parabola \(y = ax^2 + bx + c\) can be found using the formula \((h, k) = (-\frac{b}{2a}, f(h))\). Using the given function, \(a = -1\) and \(b = -18\), the vertex is \(h = -\frac{-18}{2(-1)} = -9\). Evaluate \(f(-9)\) to find \(k\): \(y = -(-9)^2 - 18(-9) - 80 = 1\). Thus, the vertex is correct at \((-9, 1)\).

Step 3: Confirm symmetry

A parabola that is symmetric with respect to a vertical line will have this line as its axis of symmetry. Since the axis of symmetry is \(x = -9\), which passes through the vertex \((-9,1)\), it confirms symmetry as an expected property given the vertex.

Step 4: Check if the point \((-8,0)\) lies on the parabola

Substitute \((-8,0)\) into the equation to verify if it satisfies it: \(-(8)^2 - 18(-8) - 80 = 0\). Calculate: \(-64 + 144 - 80 = 0\). Since this equation holds, the point \((-8,0)\) lies on the parabola.

Key Concepts

Vertex Form

The vertex form of a parabola makes it easy to identify the vertex coordinates of the function. Vertex form is written as \( y = a(x-h)^2 + k \), where \( (h, k) \) represents the vertex point of the parabola. This format is especially useful because it tells us at a glance the highest or lowest point on the parabola, known as the vertex.

In this exercise, the original quadratic was in the form \( y = -x^2 - 18x - 80 \). By transforming this into something resembling the vertex formula, we identify the vertex at \((-9, 1)\).

Reasons it's beneficial for a student to understand:

• Quickly find the vertex, which is crucial for graphing.
• Understand shifts and transformations of parabolas.

Knowing the vertex helps predict and graph the behavior of the parabola.

Axis of Symmetry

The axis of symmetry in a parabola is a vertical line that divides the parabola into two mirror images. This line passes through the vertex of the parabola. Its equation is always in the form \(x = h\), where \(h\) is the x-coordinate of the vertex.

For the given function, we calculated that \(h = -9\). Thus, the axis of symmetry is \(x = -9\).

Importance of the axis of symmetry includes:

• It helps confirm the vertex location.
• It aids in graphing the parabola accurately.

By understanding this concept, one can easily determine how the parabola will open and ensure that any plotted points reflect the correct side.

Quadratic Equation

Quadratic equations form the basis for understanding parabolas. A quadratic equation is any polynomial equation of the form \( ax^2 + bx + c = 0 \). When graphed, these equations create a shape called a parabola.

Our exercise featured the quadratic \(-x^2 - 18x - 80 = 0\). By graphing or analyzing this equation, you can tell that it represents a downward opening parabola due to the negative coefficient of \(x^2\).

The squared term, \( x^2 \), dictates that the curve will be symmetric. Understanding quadratics is vital because:

• They model many real-world scenarios.
• They set the stage for more complex algebraic concepts.

Grasping these basics allows students to explore more advanced topics in algebra and calculus.