Problem 13 - College Algebra

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=2 x^{2}-8 x+3\)

Step-by-Step Solution

Answer

The vertex of the parabola defined by the function \(f(x) = 2 x^{2} - 8 x + 3\) is at the point (2, -5).

Step 1: Identify the coefficients a and b

In the given function \(f(x) = 2 x^{2} - 8 x + 3\), the coefficients a and b are 2 and -8, respectively.

Step 2: Calculate x-coordinate of the vertex

Use the formula \(-\frac{b}{2a}\) to calculate the x-coordinate of the vertex. In our case it would be \(-\frac{-8}{2*2} = 2\).

Step 3: Calculate y-coordinate of the vertex

Now that we have the x-coordinate, let's substitute it into the function to get the y-coordinate: \(f(2) = 2*(2)^2 - 8*2 + 3 = -5\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial where the highest degree term is squared. This means the general form of a quadratic function looks like:

• \( ax^2 + bx + c \)

where \(x\) is the variable. The numbers, \(a\), \(b\), and \(c\), stand for constants, and they give the parabola its shape, size, and position.Quadratic functions are critical in algebra since they describe various real-world phenomena, from projectile motion to calculating areas. The graph of a quadratic function is a parabola, a symmetric curve, which can open either upwards or downwards based on the sign of the coefficient \(a\). If \(a > 0\), the parabola opens upward, making a U-shape. If \(a < 0\), the parabola opens downward, creating an upside-down U-shape.
Understanding these basics helps us analyze and solve various mathematical problems, such as finding the vertex, which is our next focus.

Coordinates of the Vertex

The vertex of a parabola is a very important point. It represents the highest or lowest point on the graph, depending on how the parabola opens. In simpler terms, the vertex is the tip or the pointy end of the "U" shape.To find the coordinates of the vertex for a specific quadratic function, you need both an x-coordinate and a y-coordinate.

• **X-coordinate of the Vertex:** This can be found using the formula for vertex (which we'll explain next).
• **Y-coordinate of the Vertex:** Once you have the x-coordinate, plug this value back into the original quadratic function to find the y-coordinate.

In the case of our example function, \(f(x) = 2x^2 - 8x + 3\), we've calculated:

• **X-coordinate:** 2
• **Y-coordinate:** -5

Thus, the coordinates of the vertex are \((2, -5)\). This point shows where the parabola makes its sharpest turn.

Formula for Vertex

The formula for finding the x-coordinate of the vertex is derived from the standard form of a quadratic equation. For any quadratic function \( ax^2 + bx + c \), the formula is:\[x = -\frac{b}{2a}\]This formula allows us to find the x-coordinate of the vertex quickly. The terms \(b\) and \(a\), are coefficients from the quadratic term and linear term of the function, respectively.
Once you have the x-coordinate, calculating the y-coordinate involves substituting this value back into the original equation. In our specific example, we applied the formula, \( x = -\frac{-8}{2 \times 2} = 2 \), resulting in the x-coordinate. By plugging \(x = 2\) back into the function, we found the y-coordinate, \(-5\).Thus, using the vertex formula, we determined that the vertex is located at the point \((2, -5)\), giving us crucial information about the parabola's orientation and position.