Question

Find the coordinates of the vertex and write the equation of the axis of symmetry. $$y=2 x^{2}+3 x+6$$

Short answer

The vertex of the function \(y=2 x^{2}+3 x+6\) is at (-0.75, 5.125) and the equation for the axis of symmetry is \(x=-0.75\).

Step-by-step solution

Find the Value of h (x-coordinate of the Vertex)

Use formula \(h=-b/2a\). In this case, \(a = 2\) and \(b = 3\), so \(h = -(3)/(2*2) = -0.75\)

Find the Value of k (y-coordinate of the Vertex)

Substitute \(h\) from Step 1 into the given equation to calculate \(k\). \(k = 2*(-0.75)^{2}+3*(-0.75)+6 = 5.125\)

Determine the Equation of Axis of Symmetry

The equation of the axis of symmetry is \(x=h\). So, the axis of symmetry is \(x=-0.75\).

Key concepts

Coordinates of the Vertex

Understanding the coordinates of the vertex is crucial in graphing and analyzing quadratic functions. The vertex of a quadratic equation in standard form, which is written as y = ax^2 + bx + c, is the highest or lowest point on the parabola, depending on whether the parabola opens upwards or downwards.

To find the vertex coordinates, we use the formula for the x-coordinate h = -b/(2a). Here, a and b are coefficients from the quadratic equation. Once we have h, we can plug it back into the original equation to find the y-coordinate k. In our example, the vertex coordinates are \(h, k) = (-0.75, 5.125)\). These coordinates provide important information about the function's peak and help us graph it accurately.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that passes through the vertex and divides the parabola into two mirror images. This axis is a crucial feature because it helps us understand the balance and symmetry of a parabola.

For a quadratic equation \(y = ax^2 + bx + c\), the formula to find the axis of symmetry is x = h, where h is the x-coordinate we calculated previously for the vertex. In our example, the axis of symmetry is the line \(x = -0.75\). Its equation tells us that every point on the parabola is reflected equally across this line, indicating that the vertex is not only the peak but also a central point of balance for the quadratic graph.

Quadratic Equations

Quadratic equations form the backbone of many algebraic concepts and represent parabolic graphs in coordinate geometry. They are always expressed in the form \(y = ax^2 + bx + c\), where a, b, and c are constants and the graph of these equations is a curve called a parabola.

The value of a determines the direction of the parabola's opening (upwards if a > 0 or downwards if a < 0) and its width. The axis of symmetry and the vertex help us locate the parabola on a graph and identify its key properties. Through practice, students can become proficient in manipulating these equations to reflect transformations such as shifts, stretches, and reflections of the parabola.