Problem 100 - Algebra 1

Question

Sketch the graph of the function. Label the vertex. y=\frac{1}{2} x^{2}+2 x-1

Step-by-Step Solution

Answer

The vertex of the function \(y = \frac{1}{2}x^2 + 2x - 1\) is at point (-2,-1). Graphed, this function is a parabola that opens upwards, with the vertex being the lowest point, and the y-intercept at \(y = -1\).

Step 1: Identify the quadratic function

The given function \(y = \frac{1}{2}x^2 + 2x - 1\) is a quadratic function, as it is of the form \(y = ax^2 + bx + c\). The coefficients are \(a = \frac{1}{2}\), \(b = 2\), and \(c = -1\). Quadratic functions form a U-shaped curve or a parabola when graphed.

Step 2: Calculate the Vertex

The vertex of a quadratic function in standard form \(y = ax^2 + bx + c\) is given by \((-b/2a, f(-b/2a))\). For the given function, compute the x-coordinate of the vertex as \(-b/2a = -2/(2* \frac{1}{2}) = -2\). Substitute \(-2\) into the equation to obtain the y-coordinate: \(y = \frac{1}{2}*(-2)^2 + 2*(-2) - 1 = -1\). So, the vertex is \((-2, -1)\).

Step 3: Graph the function

Start by drawing an x and y axis. Plot the vertex point that was found in the previous step. Because the coefficient of \(x^2\) is positive, the parabola opens upwards. Sketch the graph symmetrically around the vertex point. The y-intercept is found by substituting \(x = 0\) in the equation, which gives us \(y = -1\). A rough sketch of the parabola can be completed now.

Key Concepts

Vertex of a Parabola

In quadratic functions, the vertex of a parabola is a crucial component that helps us understand and graph the function. For any function in the form \(y = ax^2 + bx + c\), the vertex can be found using the formula \((-b/2a, f(-b/2a))\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation.

The x-coordinate of the vertex is calculated by the expression \(-b/2a\), which helps locate the horizontal position of the vertex on the graph. Once the x-coordinate is determined, substituting it back into the original equation gives you the y-coordinate, placing the vertex at the exact point on the Cartesian plane.

• **Why it's important:** The vertex represents the highest or lowest point of the parabola, depending on whether the parabola opens up (is U-shaped) or down (is an upside-down U).
• **Example:** For \(y = \frac{1}{2}x^2 + 2x - 1\), we found the vertex to be \((-2, -1)\). This point is essential for sketching the parabola efficiently.

Graphing Quadratics

Graphing a quadratic function involves plotting the parabola represented by the equation \(y = ax^2 + bx + c\). Here’s a step-by-step approach to sketch the graph:

First, identify the vertex using the method described previously. For example, in \(y = \frac{1}{2}x^2 + 2x - 1\), the vertex is located at \((-2, -1)\).

Once the vertex is plotted, check the direction of the parabola. When \(a > 0\), the parabola opens upwards, forming a U-shape. If \(a < 0\), it opens downwards.

• **Find additional points:** Calculate a few more points on either side of the vertex to shape the curve accurately. Use easy values such as \(x = 0\) for simple calculations.
• **Plot the y-intercept:** This is where the parabola crosses the y-axis. For our function, substituting \(x = 0\) gives \(y = -1\).

These additional points help create a symmetric and balanced parabola around the vertex. Drawing a smooth curve through these points should give a clear graph of the quadratic function.

Parabolic Shapes

The parabolic shape is a distinctive characteristic of quadratic functions, which graph into a curve called a parabola. Recognizing these shapes is key to understanding the behavior of quadratic functions in mathematics.

Parabolas can take different orientations based on the leading coefficient \(a\):

• **Upward Opening:** If \(a > 0\), the parabola opens upwards, resembling a smile. The vertex is the minimum point and represents the least y-value on the graph.
• **Downward Opening:** If \(a < 0\), the parabola opens downwards, looking like a frown. Here, the vertex is the maximum point, indicating the greatest y-value.

These shapes help in visually analyzing and determining the solutions (roots) of quadratic equations, as they often intersect the x-axis at points which are the roots of the equation.
Understanding the parabolic nature of these graphs also aids in solving many real-world problems involving area, revenue, speed, and other measurable factors where the parabola naturally arises.