Problem 28 - 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. $$ h(x)=3-4 x-4 x^{2} $$

Step-by-Step Solution

Answer

(a) The quadratic function in standard form is \( h(x) = -4x^2 - 4x + 3 \). (b) The graph is a downward-opening parabola. (c) Its maximum value occurs at the vertex.

Step 1: Rearrange into Standard Form

The standard form of a quadratic function is given by \( ax^2 + bx + c \). The function \( h(x) = 3 - 4x - 4x^2 \) can be rearranged to match this form. Reorder the terms to get \( h(x) = -4x^2 - 4x + 3 \).

Step 2: Identify the Vertex Form

To find the max or min, we convert \( h(x) = -4x^2 - 4x + 3 \) into the vertex form \( h(x) = a(x-h)^2 + k \) by completing the square.Start by factoring out \(-4\) from the first two terms:\( h(x) = -4(x^2 + x) + 3 \).Complete the square inside the bracket: ...

Key Concepts

Standard Form

The standard form of a quadratic function is the most basic and widely used way to express quadratic equations. It's important to understand this form as it helps in identifying key features of the quadratic equation. A quadratic function in standard form is written as:

• \( ax^2 + bx + c \)

where:

• \( a \) is the coefficient of \( x^2 \), the quadratic term.
• \( b \) is the coefficient of \( x \), the linear term.
• \( c \) is the constant term.

To rearrange the quadratic function from the given exercise into the standard form, the terms should be in descending order of power. For instance, in the function \( h(x) = 3 - 4x - 4x^2 \), rearrange the terms to get:
\[ h(x) = -4x^2 - 4x + 3 \]This form helps us easily identify the coefficients that influence the graph's direction, shape, and position. Here, \( a = -4 \), \( b = -4 \), and \( c = 3 \). The value of \( a \) tells us that the parabola opens downward since it's negative. Studying the standard form is crucial in understanding how the coefficients affect the shape and position of the graph.

Vertex Form

The vertex form of a quadratic function is particularly useful when you need to find the maximum or minimum point of the graph, which is known as the vertex. The vertex form is expressed as:

• \( a(x-h)^2 + k \)

Here:

• \( (h, k) \) denotes the vertex of the parabola.
• \( a \) shows the direction and width of the parabola.

Converting to vertex form often makes it easier to sketch a graph and identify the maximum or minimum value of a quadratic equation. In the exercise, we start with:
\[ h(x) = -4x^2 - 4x + 3 \]To convert into vertex form, we complete the square. Factor out the \(-4\) from the \(x^2\) and \(x\) terms:
\[ -4(x^2 + x) + 3 \]Then, complete the square within the bracket. Remember, the vertex form reveals the vertex directly, which helps in understanding where the parabola reaches its highest or lowest point.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into vertex form, making it easier to analyze its properties, especially the vertex's position. The process involves transforming part of the quadratic expression into a perfect square trinomial.

• Start by focusing on the quadratic and linear coefficients: \( x^2 + bx \).
• The goal is to add and subtract the same constant to make a perfect square trinomial.

Let's look at the standard form:
\[ h(x) = -4(x^2 + x) + 3 \]To complete the square, consider the expression inside the parenthesis:
\(x^2 + x\), take half the coefficient of \(x\), square it, and add inside:

• \[ x^2 + x = (x^2 + x + \frac{1}{4}) - \frac{1}{4} \]

Multiply the added term \( \frac{1}{4} \) by \(-4\) to adjust:

• Add the constant \(+3\) outside to balance the equation.

This transformation results in a perfect square trinomial within the quadratic expression, thus providing the parabola's vertex and easing the sketching of its graph. By mastering this method, you can gain deeper insight into the quadratic's behavior.