Question

What is the minimum value of the expression \(2 x^{2}-20 x+17 ?\)

Short answer

The minimum value of the expression \(2x^2 - 20x + 17\) is -33, which occurs at \(x = 5\).

Step-by-step solution

Identify the coefficients of the function

The coefficients of the quadratic function \(2x^2 - 20x + 17\) are \(a = 2\), \(b = -20\), and \(c = 17\).

Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola is given by the formula \(-\frac{b}{2a}\). Plugging in the given coefficients, the x-coordinate would be \(-\frac{-20}{2*2} = 5\).

Substitute the x-coordinate back into the function to get the minimum value

To obtain the minimum value, which is the y-coordinate of the vertex, we substitute the x-coordinate, 5, back into the given function. \(f(5) = 2*5^2 - 20*5 + 17 = 50 - 100 + 17 = -33\).

Interpret the result

The negative result indicates that the minimum value of the expression \(2x^2 - 20x + 17\) is -33, and it occurs at \(x = 5\). This means the vertex of the function is at the point \((5, -33)\), and the graph of this function will reach its lowest point, -33, when \(x\) is equal to 5.

Key concepts

Minimum Value

In quadratic functions, the minimum value is an important attribute. It tells us the lowest point on the graph of the function. The minimum value occurs at the vertex of the parabola associated with a quadratic function. In mathematical terms, if the parabola opens upwards, like a smile, the vertex represents the minimum value of the function.

To find this minimum value, you need to determine the y-coordinate of the vertex. Once you know the x-coordinate of the vertex, which is derived from the vertex formula, substituting it back into the original quadratic equation will give you the minimum value.

• The minimum value corresponds to the lowest point on the graph.
• For the quadratic function in the exercise, the minimum value is -33.

Hence, finding the minimum value helps in understanding where the function reaches its lowest point on the graph.

Quadratic Functions

Quadratic functions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These functions describe parabolas on a graph. Depending on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards and has a minimum value.
  
• If \(a < 0\), the parabola opens downwards and has a maximum value.

The beauty of quadratic functions lies in their symmetry around their vertex. The shape they form is a parabola, which is evenly balanced on either side of this vertex.

In practical scenarios, quadratics are often used to model situations involving area, projectile motion, and optimization problems. In the exercise provided, the function \(2x^2 - 20x + 17\) represents an upward-opening parabola.

Vertex Formula

The vertex formula is a handy tool to quickly find the vertex of a quadratic function, which is the key to determining either the minimum or maximum value. For a given quadratic function \(ax^2 + bx + c\), the x-coordinate of the vertex is calculated using the formula: \(-\frac{b}{2a}\).

This formula provides a systematic way to find the x-coordinate where the function's graph changes its direction. Once the x-coordinate of the vertex is identified, you can find the corresponding y-coordinate by plugging it back into the original equation.

• For example, in the expression \(2x^2 - 20x + 17\), \(a = 2\) and \(b = -20\), making the x-coordinate of the vertex 5.
• The vertex thus is the point \((5, -33)\).

Understanding the vertex formula not only simplifies finding the minimum or maximum values but also gives insights into the symmetry of the quadratic function.