Question

Write the quadratic function in vertex form. Then identify the vertex. \(g(x)=x^2-4 x-1\)

Short answer

The quadratic function in vertex form is \(g(x) = (x - 2)^2 - 5\), and the vertex is (2, -5).

Step-by-step solution

Write g(x) in the Form of (x - h)^2

To do this, you need to 'complete the square' for the function. For the quadratic function \(g(x) = x^2 - 4x - 1\), the coefficient of \(x\) is \(-4\), so divide it by 2 and square the result \((-4/2)^2 = 4\). Now, write \(g(x)\) in the form of \( (x - h)^2 - 4\). This yields \(g(x) = (x^2 - 4x + 4) - 4 - 1 = (x - 2)^2 - 5\).

Identify the Vertex (h,k) of the Function

The vertex form of the quadratic function is \(y = a(x - h)^2 + k\), where (h, k) is the vertex of the parabola. After completing the square, you obtained \(g(x) = (x - 2)^2 - 5\). Hence, the vertex (h, k) of the function is (2, -5).

Key concepts

Quadratic Function

A quadratic function is a specific type of polynomial function where the highest degree of the variable is 2. It’s represented in the form: \[g(x) = ax^2 + bx + c\]where:

• \(a, b,\) and \(c\) are constants, with \(a eq 0\) to ensure the quadratic nature.
• \(x\) represents the variable.

Quadratic functions graph as parabolas, which are U-shaped curves. The parabola may open upwards or downwards, depending on the sign of \(a\):

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

Knowing this, we can visualize how the function will appear on a coordinate plane. In our example function, \(g(x) = x^2 - 4x - 1\), the coefficient \(a = 1\), indicating the parabola opens upwards.

Completing the Square

Completing the square is a mathematical process used to convert a quadratic function from its standard form \(ax^2 + bx + c\) into the vertex form \(a(x-h)^2 + k\). This is especially handy because it directly reveals the vertex of the parabola. To complete the square for a given quadratic:

• Firstly, focus on the \(x^2\) and \(x\) terms: \(x^2 - 4x\).
• Take the coefficient of \(x\), divide by 2, and square it: \((-4/2)^2 = 4\).
• Add and subtract this square inside the function: \((x^2 - 4x + 4 - 4) = (x - 2)^2 - 4\).

Remember to balance the additions in the equation to maintain equality. Substituting these changes into the initial quadratic gives us \(g(x) = (x - 2)^2 - 5\), showing the function in vertex form. This completed square reveals significant details of the quadratic function, leading us to easily identify the vertex and the shape of the parabola.

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents the top or bottom of the curve. It gives important information about the function, such as the minimum or maximum value of a quadratic function, depending on the direction the parabola opens. The vertex form of a quadratic function is particularly useful:\[y = a(x - h)^2 + k\]In this form,:

• \((h, k)\) is the vertex of the parabola.
• \(h\) represents the x-coordinate of the vertex, that is, the axis of symmetry of the parabola.
• \(k\) represents the y-coordinate, which is the minimum or maximum value of the function.

From our completed form \(g(x) = (x - 2)^2 - 5\), we see that:

• The vertex \((h, k)\) is \((2, -5)\).
• This tells us the parabola opens upwards (since \(a\) is positive) with a minimum point at \((2, -5)\).

Finding the vertex helps in sketching the graph and understanding the function's behavior. It allows us to predict how the graph will look and its intersection points with the axis.