Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. $$f(x)=4 x^{2}+2 x-12$$

Step-by-Step Solution

Answer

The vertex of the function \(f(x)=4 x^{2}+2 x-12\) is (-0.25,-11.75).

Step 1: Identify the coefficients a, b, and c in the function

In the function \(f(x)=4x^2+2x-12\), the coefficients are a=4, b=2, and c=-12.

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola \(y = ax^2 + bx + c\) is given by the formula \(-b/2a\). Substitute a=4 and b=2 in the formula to get: \[x_{vertex} = -b/2a = -2/(2*4) = -0.25\]

Step 3:Calculate the y-coordinate of the vertex

The y-coordinate of the vertex is simply the value of f(x) at \(x_{vertex}\). Substituting \(x=-0.25\) into the function \(f(x) = 4x^2 + 2x - 12\) yields: \[y_{vertex} = f(-0.25) = 4*(-0.25)^2 + 2*(-0.25) - 12 = -11.75\]

Step 4: Identify the vertex

The vertex of the function is the point \((x_{vertex}, y_{vertex})\), which is \((-0.25, -11.75)\)

Key Concepts

Understanding Quadratic Functions

A quadratic function is one of the simplest types of polynomial functions and always has the form \(f(x) = ax^2 + bx + c\). The coefficients \(a\), \(b\), and \(c\) dictate the shape and position of the parabola when graphed. These functions are called quadratic because the highest exponent of the variable \(x\) is 2. This square term \(ax^2\) is what makes quadratic functions unique compared to linear functions.
When you look at a quadratic function:

• \(a\) determines if the parabola opens up or down. If \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.
• \(b\) influences the x-coordinate of the vertex.
• \(c\) is the y-intercept where the parabola crosses the y-axis.

Quadratics can model various real-world scenarios such as projectile motion or the shape of satellite dishes. Understanding each part helps you make predictions about the parabola's behavior and properties.

Graphing Parabolas Made Simple

Graphing a parabola involves understanding its various components to sketch the shape accurately. Parabolas can seem tricky at first, but by focusing on key points like the vertex and intercepts, they become easier to handle.
The graph of a quadratic function is a U-shaped curve called a parabola. Follow these steps to graph it:

• Start by locating the vertex, which is the highest or lowest point on the graph depending on whether the parabola opens upwards or downwards.
• Check the direction of the parabola by the sign of \(a\). Positive \(a\) values mean the parabola opens upward, negative values mean downward.
• Find the y-intercept by looking at the constant term \(c\), which gives you a point on the graph where it crosses the y-axis.
• Calculate the x-intercepts by setting \(f(x) = 0\) and solving for \(x\), if they exist.

By connecting these points smoothly, you'll have an accurate graph of the parabola. Understanding these elements can provide insights into the function's characteristics and help predict its behavior in different scenarios.

Mastering the Vertex Formula

The vertex formula is a handy tool for pinpointing the vertex of a parabola represented by the quadratic function \(f(x) = ax^2 + bx + c\). The vertex is a crucial component as it indicates the peak or trough of the parabola.
Use the vertex formula to find the x-coordinate of the vertex: \(x_{vertex} = -\frac{b}{2a}\). This formula tells us where the parabola's turning point is located on the horizontal axis. Once the x-coordinate is found, substitute it back into the original equation to calculate the y-coordinate, giving you the complete vertex \((x_{vertex}, y_{vertex})\).
For instance, in the function \(f(x) = 4x^2 + 2x - 12\), the vertex can be calculated as:

• x-coordinate: \(-\frac{2}{2 \times 4} = -0.25\)
• y-coordinate: substitute \(x = -0.25\) into the function to get \(y = -11.75\)
• Vertex of the parabola \((-0.25, -11.75)\)

By mastering the vertex formula, you can quickly determine this significant point in any quadratic equation, simplifying both graphing and analysis.