Problem 62 - Algebra 2

Question

Find the value of the maximum or minimum of each quadratic function to the nearest hundredth. $$ f(x)=7 x^{2}+4 x+1 $$

Step-by-Step Solution

Answer

The minimum value is approximately 0.43.

Step 1: Identify the form of the quadratic function

The given quadratic function is in the standard form, which is \( f(x) = ax^2 + bx + c \). Here, \( a = 7 \), \( b = 4 \), and \( c = 1 \).

Step 2: Determine the direction of the parabola

A quadratic function \( ax^2 + bx + c \) opens upwards if \( a > 0 \) and downwards if \( a < 0 \). Since \( a = 7 > 0 \), the parabola opens upwards, indicating that the function has a minimum value.

Step 3: Find the vertex of the quadratic function

The vertex form of a quadratic function \( ax^2 + bx + c \) can be found using the vertex formula \( x = \frac{-b}{2a} \). Substitute \( a = 7 \) and \( b = 4 \) into the formula: \( x = \frac{-4}{2 \times 7} = \frac{-4}{14} = -\frac{2}{7} \).

Step 4: Calculate the corresponding function value

Substitute \( x = -\frac{2}{7} \) back into the quadratic function to find the minimum value: \[f\left(-\frac{2}{7}\right) = 7\left(-\frac{2}{7}\right)^2 + 4\left(-\frac{2}{7}\right) + 1\]Calculate each term:\[7\left(\frac{4}{49}\right) = \frac{28}{49}, \quad 4\left(-\frac{2}{7}\right) = -\frac{8}{7}, \quad 1 = 1\]Combine and simplify:\[f = \frac{28}{49} - \frac{56}{49} + \frac{49}{49} = \frac{21}{49} = \frac{3}{7} \]Convert \( \frac{3}{7} \) to a decimal form: \( \frac{3}{7} \approx 0.43 \).

Step 5: Final Step: Round to the nearest hundredth

The minimum value of the function \( f(x) = 7x^2 + 4x + 1 \) is approximately \( 0.43 \) when rounded to the nearest hundredth.

Key Concepts

Vertex Formula

The vertex formula is a crucial tool when working with quadratic functions. It provides a method to find the vertex of the parabola, the point which is either the maximum or minimum depending on the direction it opens.
For any quadratic function in the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be determined using the formula:

• \( x = \frac{-b}{2a} \)

In this formula, the coefficients \( a \) and \( b \) are taken directly from the quadratic equation.
This formula is derived from the process of completing the square, and it tells us the axis of symmetry of the parabola.
In the example given, \( a = 7 \) and \( b = 4 \), so plugging these into the vertex formula gives us \( x = \frac{-4}{2 \times 7} = -\frac{2}{7} \). This means the parabola is symmetric around this line. After finding \( x \), substitute it back into the function to find the y-coordinate of the vertex.

Parabola

A parabola is the graph of a quadratic function, recognized by its unique U-shape. Parabolas have some interesting properties that are important for understanding quadratic functions.

• They can open either upwards or downwards.
• Their shape is determined by the coefficient \( a \).

If \( a > 0 \), the parabola opens upwards and resembles a U-shape. If \( a < 0 \), it opens downwards, resembling an upside-down U. The width of the parabola also depends on \( a \); larger values of \( |a| \) make the parabola narrower, while smaller values widen it.
The vertex of the parabola is a key point; it is the highest or lowest point on its graph, depending on the direction. In context with our function, since \( a = 7 \) is positive, the parabola opens upwards, and thus it has a minimum point at the vertex.

Standard Form of a Quadratic Function

The standard form of a quadratic function is written as \( f(x) = ax^2 + bx + c \).
This is a general form where \( a \), \( b \), and \( c \) are constants, and importantly, \( a eq 0 \). This form gives us a clear view of the parabola's orientation and sense through the following:

• Coefficient \( a \) determines the direction and width of the parabola.
• Coefficient \( b \) impacts the axis of symmetry.
• The constant \( c \) gives the y-intercept, indicating where the parabola crosses the y-axis.

To solve problems related to quadratic functions, it's often helpful to convert them into this form.
In this specific example, the function was already provided in standard form, \( f(x) = 7x^2 + 4x + 1 \), which facilitated the process of finding both the vertex and the minimum value of the quadratic function.