Question

Graph the quadratic equation. Label the vertex and axis of symmetry.\(y=2 x^2-x+3\)

Short answer

The vertex of the quadratic equation \(y=2x^2 - x + 3\) is at (0.5, 2.5), and the axis of symmetry is the line \(x = 0.5\).

Step-by-step solution

Identifying the coefficients

The given quadratic function is \(y=2x^2 - x + 3\). So, the coefficients of this quadratic equation are \(a = 2\), \(b = -1\), and \(c = 3\).

Calculating the vertex

We first need to find 'h', which is \(-b/2a\). Substituting the values of 'a' and 'b' in this formula, we get \(-(-1)/2*2=1/2 = 0.5\). To get 'k', we substitute 'h' into the equation, which gives us \(k = 2*(0.5)^2 - 0.5 + 3 = 2.5\). Therefore, the vertex (h, k) of the equation is (0.5, 2.5).

Identifying the axis of symmetry

The axis of symmetry for a quadratic function is the vertical line \(x = h\). Therefore, the axis of symmetry for this function is \(x = 0.5\).

Graphing the equation

Plot the vertex point (0.5, 2.5), draw the axis of symmetry (x = 0.5), and then sketch the parabolic curve of the quadratic function. The graph opens upwards because the coefficient 'a' is positive. The vertex is the minimum point in this case. Draw the corresponding points on both sides of the axis of symmetry in accordance to the shape of a parabola.

Key concepts

Quadratic Function Vertex

The vertex of a quadratic function is a crucial point on its graph. It represents the highest or lowest point on the parabola, which is the curve formed by graphing a quadratic equation. To find the vertex, we use the formula \(h = -\frac{b}{2a}\), where \(h\) and \(k\) are the x- and y-coordinates of the vertex respectively.

For the equation \(y = 2x^2 - x + 3\), our coefficients \(a\) and \(b\) from \(ax^2 + bx + c\) give us \(a = 2\) and \(b = -1\). Inserting these values into our formula yields \(h = 0.5\). To find \(k\), we substitute \(h\) back into the equation, resulting in \(k = 2.5\). Thus, the vertex of our parabola is \( (0.5, 2.5) \).

This point is not only a visual centerpiece on the graph but also a reflection point for the parabola, showing symmetry in the quadratic function.

Axis of Symmetry

In quadratic functions, the axis of symmetry is a vertical line that divides the parabola into two mirror images. It always passes through the vertex of the parabola. The standard form of the axis of symmetry can be represented by the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.

In our example with \(y = 2x^2 - x + 3\), we've already calculated the vertex at \( (0.5, 2.5) \), so the axis of symmetry is the line \(x = 0.5\). This line is critically important when graphing because it helps us plot points on one side of the parabola and then mirror them on the other side to ensure the curve is accurate and symmetrical.

Parabolic Curve Equations

Parabolic curve equations refer to the set of points that satisfy a quadratic function. These equations plot a symmetrical curve known as a parabola, which can either open upwards or downwards. The general form of a quadratic function is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients that determine the shape and position of the parabola.

When graphing \(y = 2x^2 - x + 3\), we notice that because \(a > 0\), the parabola opens upwards, and its vertex \( (0.5, 2.5) \) serves as the lowest point. To sketch the parabola, plot the vertex and additional points selected based on the equation, then draw a smooth curve through these points, ensuring the graph is symmetric with respect to the axis of symmetry \(x = 0.5\).

Quadratic Coefficients

Quadratic coefficients are the numerical factors in the terms of the quadratic equation. These coefficients—\(a\), \(b\), and \(c\)—greatly influence the graph's characteristics, such as the opening direction, width, and position of its vertex. In the equation \(y = ax^2 + bx + c\), \(a\) affects the parabola's direction and width; \(b\) impacts the location of the axis of symmetry; \(c\) indicates the y-intercept.

For \(y = 2x^2 - x + 3\), the positive \(a = 2\) tells us the parabola opens upward, and a larger absolute value of \(a\) makes the parabola narrower. The value of \(b = -1\) affects the slope of the parabola at the y-intercept, and \(c = 3\) is where the graph crosses the y-axis. Understanding these coefficients allows us to predict and graph the shape of the parabolic curve accurately.