Problem 34 - Algebra for College Students

Question

Find the vertex for the graph of each quadratic function. $$f(x)=-3 x^{2}+18 x-7$$

Step-by-Step Solution

Answer

(3, 20)

Step 1: Identify coefficients

A quadratic function is generally written as \[ f(x) = ax^2 + bx + c \] Identify the coefficients for the given quadratic function \[ f(x) = -3x^2 + 18x - 7 \] Here, \( a = -3 \), \( b = 18 \), \( c = -7 \)

Step 2: Calculate the x-coordinate of the vertex

Use the formula for finding the x-coordinate of the vertex of a quadratic function, which is \[ x = -\frac{b}{2a} \] Substitute the values of \(a\) and \(b\) into the formula:\[ x = -\frac{18}{2(-3)} \] Calculate the value:\[ x = -\frac{18}{-6} \] \[ x = 3 \]

Step 3: Calculate the y-coordinate of the vertex

Substitute the x-coordinate back into the original quadratic function to find the y-coordinate.\[ f(3) = -3(3)^2 + 18(3) - 7 \] Calculate the value:\[ f(3) = -3(9) + 54 - 7 \] \[ f(3) = -27 + 54 - 7 \] \[ f(3) = 20 \]

Step 4: State the vertex

The vertex of the quadratic function is the point \( (3, 20) \)

Key Concepts

quadratic function

A quadratic function is a type of polynomial function of degree 2, typically written in the form:

• \[ f(x) = ax^2 + bx + c \]

Here, \( x \) represents the variable, and \( a \), \( b \), and \( c \) are coefficients.
Quadratic functions create a U-shaped graph called a parabola. These parabolas can either open upwards (when a \( > 0 \)) or downwards (when a \( < 0 \)).
The highest or lowest point of this parabola is called the vertex.Understanding quadratic functions is essential as they appear frequently in various mathematical and real-world problems.

vertex calculation

The vertex of a quadratic function provides valuable information, as it represents the peak or the lowest point of the parabola.
To find the vertex, you need both the x-coordinate and the y-coordinate of the quadratic function's turning point.

• **X-coordinate:**the x-coordinate of the vertex can be calculated using the formula:\[ x = -\frac{b}{2a} \]Here, \( a \) and \( b \) are coefficients from the quadratic function \( f(x) = ax^2 + bx + c \).In the given problem, these coefficients are found to be \( a = -3 \) and \( b = 18 \).Plugging these into the formula gives:\[ x = -\frac{18}{2(-3)} = 3 \]
• **Y-coordinate:**To find the y-coordinate, substitute the x-value back into the original function. For our example:\[ f(3) = -3(3)^2 + 18(3) - 7 \]which simplifies to:\[ -27 + 54 - 7 = 20 \]Therefore, the y-coordinate of the vertex is 20.

Hence, the vertex of the quadratic function \( f(x) = -3x^2 + 18x - 7 \) is \( (3, 20) \).

coefficients

In a quadratic function \( f(x) = ax^2 + bx + c \), the coefficients \( a \), \( b \), and \( c \) play crucial roles in determining the shape and properties of the parabola.

• **Coefficient \( a \):**This coefficient affects the direction and the width of the parabola. If \( a \) is positive, the parabola opens upward. If \( a \) is negative, the parabola opens downward.Additionally, if \( |a| \) is large, the parabola will be narrower, and if \( |a| \) is small, the parabola will be wider.
• **Coefficient \( b \):**This coefficient affects the position of the vertex along the x-axis. It influences the slope of the parabola at the vertex.
• **Coefficient \( c \):**This coefficient represents the y-intercept of the parabola. In our example \( f(x) = -3x^2 + 18x - 7 \), \( c = -7 \), which indicates that the graph of the quadratic function will cross the y-axis at -7.

Understanding these coefficients helps in visualizing and graphing quadratic functions efficiently.