Question

Graph the quadratic equation. Label the vertex and axis of symmetry.\(y=0.5 x^2+2 x+5\)

Short answer

The vertex of the quadratic equation \(y=0.5x^2+2x+5\) is (-2, 3) and the axis of symmetry is \(x = -2\).

Step-by-step solution

Identify 'a' and 'b'

From the quadratic equation \(y=0.5x^2+2x+5\), 'a' is the coefficient of \(x^2\), which is 0.5, and 'b' is the coefficient of 'x', which is 2.

Calculate 'x' coordinate of the vertex

Use the formula for the x-coordinate of the vertex of a quadratic function, \(x = -b/2a\). Substituting the values into the formula gives \(x = -2/(2*0.5)= -2/1 = -2\).

Calculate 'y' coordinate of the vertex

Substitute 'x' in the function \(y=0.5x^2+2x+5\) to find 'y' coordinate of the vertex. This gives us \(y=0.5*(-2)^2+2*(-2)+5 = 2 - 4 + 5 = 3\). Therefore, the vertex is (-2, 3).

Identify the Axis of Symmetry

The axis of symmetry of a quadratic function is always \(x = -b/2a\). So in this case, it's \(x = -2\).

Graph the quadratic equation

Now, that we have the vertex and the axis of symmetry, we can graph the quadratic equation. Plot the vertex, draw the axis of symmetry and then sketch the graph. The graph will open upwards since 'a' is positive.

Key concepts

Vertex of a Quadratic

In the world of algebra, the vertex of a quadratic equation is a crucial concept. It represents the highest or lowest point on the graph of a quadratic function, depending on whether the parabola opens upwards or downwards, respectively. To find the vertex, you need to know the quadratic function's coefficients.

As we saw in the exercise, the equation given was y=0.5x^2+2x+5. We calculated the x-coordinate of the vertex using the formula x = -b/2a. Once we had the x-coordinate, -2, it was substituted back into the equation to find the matching y-coordinate, which turned out to be 3. Thus, the vertex for our quadratic equation is (-2, 3). Knowing the vertex is particularly useful when graphing the quadratic equation, as it gives a starting point for sketching the curve.

Axis of Symmetry

The axis of symmetry is an imaginary line that divides a parabola into mirror images, making each side a reflection of the other. For any quadratic function expressed in standard form as ax^2 + bx + c, the formula for the axis of symmetry is derived from the vertex formula and is given by x = -b/2a.

This line is paramount because it not only helps in graphing the parabola but also in solving quadratic equations and understanding their properties. In our example, by using the coefficients 'a' and 'b' from our equation y=0.5x^2+2x+5, the axis of symmetry is determined to be x = -2. This is also the x-coordinate of the vertex, reinforcing the intimate relationship between these two concepts.

Quadratic Function

A quadratic function is one of the most fundamental objects studied in algebra. It represents any polynomial equation of the second degree, generally written in the form y=ax^2+bx+c. The graph of a quadratic function is a curve called a parabola. These parabolas can either open upwards or downwards which is determined by the sign of the coefficient 'a'.

If 'a' is positive, as seen in our example function y=0.5x^2+2x+5, the graph opens upwards, and we have a minimum point at the vertex. If 'a' were negative, the graph would open downwards, and the vertex would represent a maximum point. The quadratic function is significant in both theoretical and practical applications across various fields such as physics, engineering, and economics.

Coefficient Calculations

• The coefficients of a quadratic equation are the numerical factors that multiply each term. In the equation y=0.5x^2+2x+5, 'a' is the coefficient of the x^2 term, 'b' is the coefficient of the 'x' term, and 'c' is the constant term.
• Understanding how to identify and calculate these coefficients is important for graphing the equation, determining the vertex and the axis of symmetry, and solving the function.
• The primary coefficients we focused on for the purpose of graphing the parabola are 'a' and 'b'. The 'c' coefficient affects the vertical position of the parabola but does not directly factor into the calculation of the vertex or axis of symmetry.

Accurate coefficient calculations lead to a correct representation of the quadratic function's graph. Simplifying coefficients, when possible, can make these calculations and subsequent steps of finding the vertex and axis of symmetry simpler.