Problem 12 - Algebra 2

Question

Identify the conic section represented by each equation by writing the equation in standard form. For a parabola, give the vertex. For a circle, give the center and the radius. For an ellipse or a hyperbola, give the center and the foci. Sketch the graph. \(x^{2}-8 x-y+19=0\)

Step-by-Step Solution

Answer

The equation represents a parabola with the vertex at (4, 3).

Step 1: Identify the conic section type

Here, the given equation is \(x^{2}-8x-y+19=0\). It seems to fit the standard form of a parabola, namely \(y=ax^{2}+bx+c\), because it contains a x^2 term, but no corresponding y^2 term.

Step 2: Write the equation in standard form

To check our assumption and to find the definite form we rewrite the given equation in the standard form of a parabola, which involves completing the square. We bring the -y term on the other side and then complete the square with x terms: \[ y = x^{2} - 8x + 19 \] \[ y = (x^{2} - 8x + 16) - 16 + 19 \] \[ y = (x - 4) ^{2} + 3 \] . We have the standard form \(y=a(x-h)^{2} + k\), where h and k are the x and y coordinates of the vertex.

Step 3: Identify the vertex

From the standard form, we can identify the vertex (h, k) as (4, 3).

Step 4: Sketch the graph

Sketching the graph of a parabola involves locating the vertex at the point (h, k) = (4, 3) and drawing a U-shaped curve, opening upward (since the coefficient of the (x - h)^2 term is positive). Note that the y-intercept is at (0,19), the graph should pass this point. Further points can be calculated by replacing x by numbers greater and smaller than 4 and calculating the corresponding y.

Key Concepts

Parabola

A parabola is a familiar shape in mathematics that resembles a U. This U-shape is symmetrical around a vertical line called the axis of symmetry. Parabolas can open upwards or downwards depending on the equation. They are part of a group known as conic sections, which also includes circles, ellipses, and hyperbolas.
Often seen in physics and engineering, parabolas appear when dealing with projectile motion and satellite dishes. In algebra, the formula for a parabola includes a squared term, typically in the form of \(y = ax^2 + bx + c\).

• The squared term indicates that the graph will form a parabolic shape.
• Each parabola has a single highest or lowest point, known as the vertex.

Standard Form

The standard form of a parabola makes it easy to identify its key characteristics, like the vertex. It provides a clear structure that helps in graph plotting and understanding the parabola’s direction.
For a vertical parabola, the standard form is given by:\[ y = a(x-h)^2 + k \]

• Here, \(a\) affects the width and direction of the parabola. A positive \(a\) means it opens upwards, while a negative \(a\) means it opens downwards.
• The point \((h, k)\) represents the vertex.

In converting quadratic equations to this standard form, terms are arranged such that the math is easier to interpret and graph.

Vertex

The vertex of a parabola is crucial for understanding the shape and position of the graph. It is the point where the parabola reaches its maximum or minimum value, depending on which way it opens.
In a quadratic equation transformed to standard form, the vertex is simply \((h, k)\). For example, in the equation \(y = (x-4)^2 + 3\), the vertex would be at \((4, 3)\).

• The vertex can be found directly from the standard form without complex calculations.
• It represents the axis of symmetry, meaning the graph is mirrored on either side of this point.

Finding the vertex is the first step in graphing parabolas, as it serves as an anchor point for the entire curve.

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into its standard form. This process helps in identifying the vertex of a parabola with ease.
To complete the square:

• Start by rearranging the quadratic equation strictly in terms of \(x\).
• Identify the coefficient of \(x\), divide it by 2, and square the result.
• Add and subtract this square term within the equation to balance it.

In the equation \(x^2 - 8x - y + 19 = 0\), completing the square involves adjusting the \(x\) part:

• Transform \(x^2 - 8x + \_\) by adding and subtracting \(16\) to get \((x - 4)^2 - 16\).
• Then, rearrange into the standard form \(y = (x-4)^2 + 3\) by adjusting the constant on the other side.

This technique modifies the structure of the equation, allowing easy identification of the vertex and facilitating graph plotting.