Problem 38 - Introductory Algebra for College Students

Question

Find the vertex for the parabola whose equation is given by first writing the equation in the form \(y=a x^{2}+b x+c\) $$y=(x-4)^{2}+3$$

Step-by-Step Solution

Answer

The vertex of the parabola given by the equation \(y=(x-4)^{2}+3\) is \((4, 3)\).

Step 1: Write the equation in standard form

The equation given is \(y = (x-4)^2+3\). To write it in standard form, we need to expand the squared term: \((x-4)^2 = x^2 - 8x + 16\). Therefore, the standard form of this equation is \(y = x^2 - 8x + 16 + 3\), or simply \(y = x^2 - 8x + 19\).

Step 2: Identify the coefficients

From the standard form identified in Step 1, we can see that the coefficient \(a = 1\), \(b = -8\), and \(c = 19\).

Step 3: Calculate the vertex of the parabola

The vertex of a parabola is given by \((h, k)\), where \(h = -b/(2a)\) and \(k = c - (b^2/(4a))\). Substituting the values identified in Step 2, we calculate \(h = 8/(2*1) = 4\) and \(k = 19 - ((-8)^2/(4*1)) = 19 - 16 = 3\). Therefore, the vertex of the parabola is \((4, 3)\).

Key Concepts

Standard Form of a Quadratic Equation

A quadratic equation is a type of polynomial equation that has the form \( y = ax^2 + bx + c \). This specific format is referred to as the "standard form" of a quadratic equation. Knowing this form is crucial because it offers a clear picture of the parabola's shape and direction, which is determined by the coefficient \( a \). The graph of a quadratic equation is always a parabola. A parabola can open upwards or downwards, directly influenced by the sign of \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

For the equation \( y = x^2 - 8x + 19 \), the coefficients are \( a=1 \), \( b=-8 \), and \( c=19 \). With these values, you can quickly identify that the parabola opens upwards because \( a = 1 > 0 \). The standard form not only lets you find the vertex but also makes it easier to perform other algebraic operations, like finding roots or factoring.

Vertex Calculation

Calculating the vertex of a parabola is a straightforward process once you have the quadratic equation in standard form. The vertex \((h, k)\) of the parabola describes its highest or lowest point, depending on whether it opens upwards or downwards. Here's how to calculate it:

• The formula for \( h \) is \( -\frac{b}{2a} \).
• The formula for \( k \) is \( c - \frac{b^2}{4a} \).

Using these formulas, let's calculate the vertex for our example equation \( y = x^2 - 8x + 19 \):

• To find \( h \): Substitute \( b = -8 \) and \( a = 1 \) into the formula: \( h = -\frac{-8}{2 \times 1} = 4 \).
• To find \( k \): Substitute \( b = -8 \), \( a = 1 \), and \( c = 19 \) into the formula: \( k = 19 - \frac{(-8)^2}{4 \times 1} = 19 - 16 = 3 \).

So, the vertex of the parabola is \((4, 3)\). This point is significant because it represents the parabola's turning point (minimum or maximum).

Quadratic Equation Coefficients

In a quadratic equation written in standard form \( y = ax^2 + bx + c \), the letters \( a \), \( b \), and \( c \) are known as coefficients. Each of these plays a different role in shaping the parabola:

• \( a \) affects the parabolas' width and direction (opens upwards if positive, downwards if negative).
• \( b \) influences the position of the vertex along the x-axis.
• \( c \) represents the y-intercept of the parabola, where it crosses the y-axis when \( x = 0 \).

For the equation \( y = x^2 - 8x + 19 \), the coefficients were identified as \( a = 1 \), \( b = -8 \), and \( c = 19 \). Understanding these coefficients allows you to gain insights into the parabola's sketch even before plotting it:

• Since \( a = 1 \) (a positive number), we expect the parabola to open upwards.
• The linear coefficient \( b = -8 \) shifts the vertex horizontally.
• The constant term \( c = 19 \) marks its point on the y-axis.

Grasping these concepts helps to predict how changes in \( a \), \( b \), and \( c \) will alter the parabola's appearance without having to necessarily graph it.