Problem 4 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=-4 x^{2}$$

Step-by-Step Solution

Answer

The function is a downward-opening parabola with the vertex at (0,0).

Step 1: Identify the Function Type

The given function is \(f(x) = -4x^2\). This is a quadratic function, as it is in the form of \(ax^2 + bx + c\). Here, \(a = -4\), \(b = 0\), and \(c = 0\). Quadratic functions graph to form a parabola.

Step 2: Determine the Vertex

For the function \(f(x) = ax^2\), the vertex is at \((0, 0)\) if it is in the form \(-4x^2\) without a linear term or constant. The vertex of the parabola for this function is at the origin, \((0, 0)\).

Step 3: Determine the Direction of the Parabola

The coefficient \(a = -4\) is negative, which means the parabola opens downwards. If \(a\) is positive, the parabola opens upwards, and if it's negative, it opens downwards.

Step 4: Find Additional Points

Choose some x-values to substitute into the function to find corresponding y-values. For example:- If \(x = 1\), then \(f(1) = -4(1)^2 = -4\).- If \(x = -1\), then \(f(-1) = -4(-1)^2 = -4\).- If \(x = 2\), then \(f(2) = -4(2)^2 = -16\).- If \(x = -2\), then \(f(-2) = -4(-2)^2 = -16\).The points \((1, -4)\), \((-1, -4)\), \((2, -16)\), and \((-2, -16)\) can be plotted.

Step 5: Sketch the Graph

Plot the vertex \((0, 0)\) and the additional points \((1, -4)\), \((-1, -4)\), \((2, -16)\), and \((-2, -16)\) on a coordinate grid. Draw a smooth curve connecting these points to form the parabola. Make sure the parabola opens downward.

Key Concepts

Understanding Quadratic Functions

A quadratic function is a special type of polynomial function. It has the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This form is known as the standard form of a quadratic function.

• **Coefficient \(a\):** This determines the direction the parabola opens. A positive \(a\) results in a parabola that opens upwards, and a negative \(a\) results in a parabola that opens downwards.
• **Terms \(b\) and \(c\):** These will affect the position of the vertex and the shape of the parabola but won’t change its general direction.

Quadratic functions are fascinating because they always graph as a parabola. This consistent shape makes them easy to recognize and work with on a graph. Understanding each part of the quadratic equation helps in sketching the characteristic curve correctly.

Exploring the Parabola

A parabola is the graph of a quadratic function, and it possesses a distinctive "U" shape. Specific properties make parabolas interesting and useful.

### Key Features of Parabolas

• **Direction of Opening:** The coefficient \(a\) in the quadratic function \(ax^2 + bx + c\) determines if the parabola opens upwards or downwards.
• **Symmetry:** Every parabola has an axis of symmetry, which is a vertical line that passes through the vertex. For the function \(f(x) = ax^2\), it is the y-axis itself.
• **Width:** The absolute value of \(a\) also influences the width of the parabola. A larger absolute value makes the parabola narrower, while a smaller absolute value makes it wider.

Understanding these features helps to graph parabolas accurately and predict their behavior from just looking at the quadratic equation.

Locating the Vertex and Its Role

The vertex of a parabola is a crucial point on its graph. It represents the highest or lowest point depending on the direction the parabola opens. For the quadratic \(f(x) = ax^2 + bx + c\), finding the vertex requires some understanding of its formula.

### Determining the VertexFor a parabola described by \(f(x) = ax^2\) like in our example \(f(x) = -4x^2\), the vertex is straightforward to find:

• If the quadratic function does not include linear or constant terms (i.e., \(b = 0\) and \(c = 0\)), the vertex is at the origin \((0, 0)\).

Since the vertex is either the highest or lowest point on a parabola, it can help to understand the whole structure of the graph. In a real-world context, the vertex could represent maximum profit or minimum cost, which is why it is essential in analysis and problem-solving.