Problem 60 - Algebra 2

Question

Find the value of the maximum or minimum of each quadratic function to the nearest hundredth. $$ f(x)=2 x^{2}-3 x+2 $$

Step-by-Step Solution

Answer

The minimum value is 0.88.

Step 1: Identifying the Coefficients

For the quadratic function \[ f(x) = ax^2 + bx + c \] where the given function is \[ f(x) = 2x^2 - 3x + 2 \] we identify: - \(a = 2\)- \(b = -3\)- \(c = 2\).

Step 2: Finding the Vertex's X-Coordinate

The x-coordinate of the vertex for a quadratic function \[ ax^2 + bx + c \] can be determined by the formula: \[ x = -\frac{b}{2a} \] Plugging in the values:\[ x = -\frac{-3}{2 \times 2} = \frac{3}{4} \] Thus, the x-coordinate of the vertex is \(0.75\).

Step 3: Calculating the Vertex's Y-Coordinate

Substitute the x-coordinate back into the function to find the y-coordinate:\[ f\left(\frac{3}{4}\right) = 2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 2 \]Calculate step-by-step:- \(\left(\frac{3}{4}\right)^2 = \frac{9}{16}\)- \(2 \times \frac{9}{16} = \frac{18}{16} = \frac{9}{8}\)- \(3 \times \frac{3}{4} = \frac{9}{4}\)Then compute:\[ \frac{9}{8} - \frac{9}{4} + 2 \]Common denominator for \( \frac{9}{4}\) and \(2\) is \(8\):- \(\frac{9}{4} = \frac{18}{8}\)- \(2 = \frac{16}{8}\)Final calculation:\[ \frac{9}{8} - \frac{18}{8} + \frac{16}{8} = \frac{7}{8} \]Thus, the y-coordinate of the vertex is \(0.875\).

Step 4: Concluding the Maximum or Minimum Value

Since the coefficient \(a = 2\) is positive, the parabola opens upwards, indicating the vertex represents a minimum value. Thus, the minimum value of the function is the y-coordinate of the vertex \(0.875\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is particularly helpful when identifying the position of a parabola. It is expressed as \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. Converting a quadratic equation to vertex form can simplify finding the vertex, which is either the highest or lowest point of the parabola based on its orientation.

Unlike the standard form \( ax^2 + bx + c \), vertex form readily gives you the vertex directly from the equation. This form is especially beneficial when sketching graphs because knowing the vertex allows you to understand the parabola's peak ( extit{vertex}) and direction.

• If \( a > 0 \), the parabola opens upwards, and the vertex is the minimum point.
• If \( a < 0 \), it opens downwards, and the vertex is the maximum point.

By finding the vertex using either method—completing the square or using the vertex formula \( x = -\frac{b}{2a} \) as in our example—you can rewrite or understand the function in its vertex form.

Minimum Value

In the context of quadratic functions, a minimum or maximum value refers to the value of the function at its vertex. This point can be determined quickly when the function is in vertex form, but can also be found from standard form using the vertex formula. In our exercise, we calculated the parabola's vertex when given \( f(x) = 2x^2 - 3x + 2 \).

We determined the x-coordinate of the vertex using \( x = -\frac{b}{2a} \), yielding \( x = 0.75 \). By substituting this x value back into the original function, we found the y-coordinate, which in this case was \( 0.875 \). This y-coordinate is the minimum value because:

• The coefficient \( a = 2 \) is positive.
• Therefore, the parabola opens upwards.
• The vertex point is the lowest point on this curve.

Thus, a positive \( a \) implies the vertex's y-coordinate is the minimum value of the function. Understanding this helps in solving optimization problems and analyzing the growth/decay situations in applications.

Coefficient Identification

Identifying coefficients in a quadratic function in standard form \( ax^2 + bx + c \) is essential for analyzing the function's characteristics and solving related problems. These coefficients dictate the shape and position of the parabola represented by the quadratic equation.

• \( a \): Determines the direction of the parabola (upwards for \( a > 0 \), downwards for \( a < 0 \)) and affects the width.
• \( b \): Impacts the location of the axis of symmetry and thus the x-coordinate of the vertex.
• \( c \): Represents the y-intercept of the parabola, where the graph intersects the y-axis.

For the exercise, we identified \( a = 2 \), \( b = -3 \), and \( c = 2 \) from \( f(x) = 2x^2 - 3x + 2 \). Recognizing these coefficients aids in converting the quadratic to vertex form, finding the vertex, and graphing or analyzing the behavior of the quadratic function. Accurate identification of these coefficients is crucial in solving quadratic problems and ensures understanding of the function's graphical representation.