Problem 15 - College Algebra

Question

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=2 x^{2}-10 x+4 $$

Step-by-Step Solution

Answer

The function has a minimum value of -8.5 at x = 2.5 with axis of symmetry x = 2.5.

Step 1: Understand the Structure of a Quadratic Function

The given quadratic function is represented in the form \(f(x) = ax^2 + bx + c\), where \(a = 2\), \(b = -10\), and \(c = 4\). In this case, it has a positive \(a\) value, which indicates a parabola that opens upwards, suggesting a minimum value.

Step 2: Find the Axis of Symmetry

The axis of symmetry for a quadratic function in the form \(ax^2 + bx + c\) is calculated using the formula \(x = -\frac{b}{2a}\). Substitute the values of \(b = -10\) and \(a = 2\) into the formula:\[x = -\frac{-10}{2(2)} = \frac{10}{4} = 2.5\]Thus, the axis of symmetry is \(x = 2.5\).

Step 3: Determine the Vertex and Minimum Value

The vertex of the parabola is at \(x = 2.5\). To find the minimum value of the function, substitute \(x = 2.5\) back into the function \(f(x)\):\[f(2.5) = 2(2.5)^2 - 10(2.5) + 4\]Calculate each term:\(2(2.5)^2 = 2(6.25) = 12.5\),\(-10(2.5) = -25\),and the constant is \(4\).Add these values:\(12.5 - 25 + 4 = -8.5\).Therefore, the minimum value of the function is \(-8.5\).

Key Concepts

Axis of Symmetry

Understanding the axis of symmetry is crucial when dealing with quadratic functions. It is a vertical line that divides the parabola into two mirror-image halves. This happens because a parabola is symmetric. For any quadratic function of the form \(ax^2 + bx + c\), you can find the axis of symmetry using the formula \(x = -\frac{b}{2a}\). In our function \(f(x) = 2x^2 - 10x + 4\), we substitute \(a = 2\) and \(b = -10\) into this formula. Doing the calculations, \(x = -\frac{-10}{2(2)} = \frac{10}{4} = 2.5\). Hence, the axis of symmetry is \(x = 2.5\). This axis not only splits the parabola into two equal halves but also passes through the vertex.

Vertex

The vertex of a parabola is the point where it reaches its highest or lowest value, serving as the point of return. It lies on the axis of symmetry. For our function, with the axis located at \(x = 2.5\), this is precisely the \(x\)-coordinate of our vertex.
To find the \(y\)-coordinate of the vertex, substitute \(x=2.5\) back into the quadratic function:
\[f(2.5) = 2(2.5)^2 - 10(2.5) + 4\] After doing the math, you get \(f(2.5) = 12.5 - 25 + 4 = -8.5\).
Therefore, the vertex is at the point \((2.5, -8.5)\). This vertex corresponds to the minimum value of the function, as seen in the next sections.

Parabolas

Parabolas are the U-shaped graphs that represent quadratic functions. The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic equation \(ax^2 + bx + c\). If \(a\) is positive, like in our function \(f(x) = 2x^2 - 10x + 4\), the parabola opens upwards. Conversely, if \(a\) is negative, the parabola would open downwards.
With an upward opening parabola, the vertex is the minimum point, while if it opens downward, the vertex would be the maximum point.

• Positive \(a\): Parabola opens upwards - Minimum value at the vertex
• Negative \(a\): Parabola opens downwards - Maximum value at the vertex

Understanding the structure of parabolas helps in predicting the kind of solutions we can expect from a quadratic function.

Minimum Value

In the context of quadratic functions, the minimum value refers to the lowest point or bottom of the parabola, where it has a positive \(a\) value. This is crucial for functions like \(f(x) = 2x^2 - 10x + 4\), as it opens upwards and has a clear minimum point. Once we locate the vertex at \((2.5, -8.5)\), we understand that this is the point of minimum value.
The minimum value of \(-8.5\) represents the smallest output the function can achieve for any real \(x\), making it significant in optimization problems or real-world scenarios where minimization of values is needed.
Always remember that for any upward opening parabola, the \(y\)-coordinate of the vertex gives you the minimum value of the function. This characteristic is practical when trying to find the optimal solution in various applications.