Problem 26 - Algebra and Trigonometry

Question

19-28 \(=\) A quadratic function is given. (a) Express the quadratic function in standard form. (b) Sketch its graph. (c) Find its maximum or minimum value. $$ g(x)=2 x^{2}+8 x+11 $$

Step-by-Step Solution

Answer

(a) Standard form: Already given as \( g(x) = 2x^2 + 8x + 11 \). (b) Graph: Opens upwards with vertex at (-2, 3). (c) Minimum value: 3.

Step 1: Identifying the Quadratic Coefficients

The given quadratic function is \( g(x) = 2x^2 + 8x + 11 \). In this function, the coefficients of the quadratic are \( a = 2 \), \( b = 8 \), and \( c = 11 \). These coefficients are essential for expressing the function in standard form, analyzing the graph, and finding its maximum or minimum value.

Step 2: Standard Form of a Quadratic Function

The standard form of a quadratic function is \( g(x) = ax^2 + bx + c \). The given equation \( g(x) = 2x^2 + 8x + 11 \) is already in standard form because it follows the general structure with \( a = 2 \), \( b = 8 \), and \( c = 11 \).

Step 3: Vertex Form Conversion to Facilitate Graphing

Convert the quadratic into vertex form, which is \( g(x) = a(x-h)^2 + k \). To do this, we complete the square: 1. Take half of \( b \) (which is 8), square it, and adjust the expression. Therefore, \( (\frac{8}{2})^2 = 16 \). 2. Rewrite the quadratic part: \( 2((x+2)^2 - 4) = 2(x+2)^2 - 8 \). 3. Adjust constant term to keep it equal: \( g(x) = 2(x+2)^2 - 8 + 11 = 2(x+2)^2 + 3 \).So, the vertex form is \( g(x) = 2(x+2)^2 + 3 \).

Step 4: Sketching the Graph of the Quadratic Function

Using the vertex form \( g(x) = 2(x+2)^2 + 3 \), note that the vertex \( (h, k) \) is \((-2, 3)\) since \(h = -2\) and \(k = 3\). Plot the vertex on a coordinate plane. Because \( a = 2 \) is positive, the parabola opens upwards. Sketch the parabola accordingly with the vertex as its lowest point.

Step 5: Finding the Minimum Value

Since the parabola opens upwards (\( a = 2 > 0 \)), it has a minimum value at its vertex. From the vertex form \( g(x) = 2(x+2)^2 + 3 \), the minimum value occurs at \( x = -2 \) and is \( g(-2) = 3 \). Therefore, the minimum value of the function is 3.

Key Concepts

Standard Form

The standard form of a quadratic function is an essential way to express these functions. It follows the format \( ax^2 + bx + c \). Here:

• \( a \) is the coefficient in front of \( x^2 \), which indicates the parabola's width and direction of opening.
• \( b \) is the linear coefficient in front of \( x \).
• \( c \) is the constant term.

The given quadratic equation \( g(x) = 2x^2 + 8x + 11 \) is already in standard form. This makes it easier to identify the quadratic nature of the equation and prepares it for further analysis or conversion, such as turning it into vertex form for graphing purposes.
Recognizing the coefficients from this form, you can perform various operations, such as completing the square, that may help in graph interpretation and function analysis.

Vertex Form

The vertex form of a quadratic function is a transformation that makes it straightforward to find the vertex of the parabola. It is written as \( g(x) = a(x-h)^2 + k \), where the vertex is at the point \((h, k)\).To convert from standard form to vertex form, complete the square:

• Identify the coefficient \( b \). For \( g(x) = 2x^2 + 8x + 11 \), this is 8.
• Take half of \( b \), divide it by 2, and square it: \( \left( \frac{8}{2} \right)^2 = 16 \).
• Add and subtract this squared value inside the equation, adjusted by the coefficient \( a \).

After completing these steps, the given equation becomes \( g(x) = 2(x+2)^2 + 3 \). The vertex of the parabola can now be easily read as \((-2, 3)\), making plotting much more intuitive.

Graphing Parabolas

Graphing a quadratic function involves understanding its shape and where it sits on the coordinate plane. Using the vertex form of a function helps immensely with this.For the equation \( g(x) = 2(x+2)^2 + 3 \), you can plot the parabola starting from its vertex at \((-2, 3)\). Note the following:

• The coefficient \( a = 2 \) is positive, so the parabola opens upwards.
• The value of \( a \) also affects the width of the parabola; a larger value makes it narrower.

Begin your graph by marking the vertex on the coordinate system. Since the parabola opens upwards, the branches extend away from the vertex
You can plot additional points by choosing values for \( x \) around the vertex and calculating corresponding \( g(x) \) values. This gives you a full understanding of the parabola's trajectory.

Maximum and Minimum Values

For quadratic functions, maximum and minimum values refer to the highest or lowest point on the graph of the function. A quick way to determine these is by looking at the vertex of the parabola.In vertex form \( g(x) = 2(x+2)^2 + 3 \):

• The value of \( a = 2 \) is positive, indicating that the parabola opens upwards. This means the graph has a minimum value.
• The minimum value occurs at the vertex, \((h, k) = (-2, 3)\).

The function \( g(x) \) attains its minimum value of 3 at \( x = -2 \). If \( a \) were negative, the parabola would open downwards, and this point would be a maximum instead. Understanding the sign and value of \( a \) is crucial in predicting these values.