Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write \(f(x)=-3 x^{2}+30 x-4\) in the form $$ f(x)=a(x-h)^{2}+k $$

Short answer

\( f(x) = -3(x - 5)^2 + 71 \)

Step-by-step solution

- Identify the Coefficients

Identify the coefficients in the given quadratic function. Here, the function is given as \( f(x) = -3x^2 + 30x - 4 \). The coefficients are: \(a = -3\), \(b = 30\), and \(c = -4\).

- Complete the Square

To write the quadratic function in the form \( f(x) = a(x-h)^2 + k \), complete the square. Start by factoring out \(a\) from the first two terms:\[ f(x) = -3(x^2 - 10x) - 4 \] Next, add and subtract the square of half the coefficient of \(x\) inside the parentheses:\[ f(x) = -3(x^2 - 10x + 25 - 25) - 4 \] Simplify inside the parentheses:\[ f(x) = -3((x - 5)^2 - 25) - 4 \]

- Simplify the Expression

Distribute the \(-3\) and combine like terms:\[ f(x) = -3(x - 5)^2 + 75 - 4 \] Combine the constants:\[ f(x) = -3(x - 5)^2 + 71 \]

Key concepts

Quadratic Functions

A quadratic function is a type of polynomial function that can be written in the general form: \ \(f(x) = ax^2 + bx + c\). These functions have graphs that form a curve called a parabola. The parameter \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)). The values of \(b\) and \(c\) influence the position and shape of the parabola.
Quadratic functions are fundamental in algebra and appear in various real-world contexts, such as physics, engineering, and economics. Understanding how to rewrite these functions in different forms makes it easier to analyze and graph them.

Standard Form

The standard form of a quadratic function is: \ \(f(x) = ax^2 + bx + c\). Here:

• \(a\) is the coefficient of \(x^2\) and it determines the direction and width of the parabola.
• \(b\) is the coefficient of \(x\) and it affects the position of the vertex along the x-axis.
• \(c\) is the constant term and it determines the y-intercept of the graph.

In our example, the quadratic function is given in standard form as: \ \(f(x) = -3x^2 + 30x - 4\). Here, \(a = -3\), \(b = 30\), and \(c = -4\). Each of these coefficients plays a key role in shaping the graph of the parabola.

Vertex Form

The vertex form of a quadratic function is: \ \(f(x) = a(x - h)^2 + k\), where \( (h, k) \) is the vertex of the parabola. The process of converting from standard form to vertex form is known as completing the square. This transformation makes it easier to identify the vertex and other key features of the graph.
For our example, the process involves:

• Factoring out the coefficient of \(x^2\): \(f(x) = -3(x^2 - 10x) - 4\)
• Adding and subtracting the square of half the coefficient of \(x\): \(f(x) = -3(x^2 - 10x + 25 - 25) - 4\)
• Rewriting the quadratic expression inside the parentheses: \(f(x) = -3((x - 5)^2 - 25) - 4\)
• Distributing the leading coefficient and simplifying: \(f(x) = -3(x - 5)^2 + 75 - 4 = -3(x - 5)^2 + 71\)

The final vertex form of the function is: \ \(f(x) = -3(x - 5)^2 + 71\). Here, the vertex of the parabola is \((5, 71)\).