Problem 25 - 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. $$ g(x)=3 x^{2}-12 x+13 $$

Step-by-Step Solution

Answer

The function is in standard form, the graph is a parabola opening upwards, and the minimum value is 1.

Step 1: Identify the Standard Form

The standard form of a quadratic function is \( ax^2 + bx + c \). For the given function \( g(x) = 3x^2 - 12x + 13 \), it's already in standard form where \( a = 3 \), \( b = -12 \), and \( c = 13 \).

Step 2: Determine the Vertex

To find the vertex, use the formula \( x = -\frac{b}{2a} \). Plug in \( b = -12 \) and \( a = 3 \): \[ x = -\frac{-12}{2 \times 3} = 2 \].Then substitute \( x = 2 \) back into \( g(x) \) to find \( y \): \[ g(2) = 3(2)^2 - 12(2) + 13 = 12 - 24 + 13 = 1 \].The vertex is \((2, 1)\).

Step 3: Sketch the Graph

The parabola opens upwards as \( a = 3 > 0 \). The vertex at \((2, 1)\) is the minimum point. Plot the vertex and a few points on each side of \( x = 2 \) to draw the parabola.

Step 4: Find the Minimum Value

Since the parabola opens upwards, the vertex represents the minimum point of the function. The minimum value of \( g(x) \) is the \( y \)-value at the vertex, which is \( g(2) = 1 \).

Key Concepts

Standard Form

The standard form of a quadratic function is one of the key foundations of understanding and solving quadratic equations. This form is expressed as \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Here, \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant term, which doesn't have a variable attached to it. In the given function \( g(x) = 3x^2 - 12x + 13 \), you can clearly identify that:

• \( a = 3 \)
• \( b = -12 \)
• \( c = 13 \)

This is a straightforward quadratic equation already presented in standard form. Remembering this form helps you easily recognize and manipulate quadratic functions for various calculations and graphing tasks.

Vertex of a Parabola

Finding the vertex of a parabola is an important step in understanding the overall behavior of a quadratic function. The vertex is essentially the 'turning point' of the parabola, where it changes direction. It's either the highest or lowest point on the graph, depending on the parabola's orientation (upwards or downwards). To find the vertex of a parabola represented by a quadratic function \( ax^2 + bx + c \), you use the formula for the \( x \)-coordinate of the vertex:\[ x = -\frac{b}{2a} \]Once you have the \( x \)-coordinate, substitute it back into the original function to find the \( y \)-coordinate. This pair \((x, y)\) gives you the vertex's precise location.For the function \( g(x) = 3x^2 - 12x + 13 \), calculate:

• \( x = -\frac{-12}{2 \times 3} = 2 \)
• Find \( g(2) = 3(2)^2 - 12 \cdot 2 + 13 = 1 \)

Therefore, the vertex is \((2, 1)\), which indicates the minimum point on this upward-opening parabola.

Minimum Value of a Function

The minimum value of a quadratic function is quite significant, as it reflects the lowest point the function can reach. In a quadratic function of the form \( ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards. This means the vertex represents the minimum value of the function. Conversely, if \( a < 0 \), the parabola opens downwards, and the vertex would instead be the maximum value.In this context, the function \( g(x) = 3x^2 - 12x + 13 \), since \( a = 3 > 0 \), opens upwards. Therefore, the minimum value of the function is found at the vertex. We've already computed the vertex to be \((2, 1)\).Thus, the minimum value of the function is simply the \( y \)-value at the vertex:

• The minimum value of \( g(x) \) is 1.

Understanding this principle is crucial when examining how a quadratic function behaves, particularly in optimization problems and real-world applications.