Problem 22 - Algebra 2

Question

Identify the vertex and the \(y\) -intercept of the graph of each function. $$ y=0.1(x-3.2)^{2} $$

Step-by-Step Solution

Answer

The vertex of the graph of the function is at \((3.2, 0)\), and the y-intercept is at \((0, 1.024)\).

Step 1: Identify the Vertex of the Parabola

The given quadratic function is in vertex form, which is written as \(y=a(x-h)^2+k\), where \((h, k)\) is the vertex of the parabola. For the given function \(y=0.1(x-3.2)^2\), the vertex \(h\) is 3.2 and since there is no \(k\) value added or subtracted from the square, \(k\) is 0. Therefore, the vertex of the function is \((3.2, 0)\).

Step 2: Find the Y-intercept

To find the \(y\)-intercept, set \(x\) to 0 and solve for \(y\). Plugging 0 into the equation, we get \(y=0.1(0-3.2)^2\). Simplifying, we find \(y=0.1(10.24)\), which gives us \(y=1.024\). The \(y\)-intercept is therefore \((0, 1.024)\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when studying quadratic functions. The vertex represents the highest or lowest point on the graph of a parabola, depending on whether the parabola opens upwards or downwards. In our example, the quadratic function given is in the vertex form, which is written as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

For the function y = 0.1(x - 3.2)^2, we can identify the vertex by looking at the values of h and k. Here, h is 3.2, and k is implicitly 0 (since there is no term added or subtracted after the squared term). This gives us a vertex at the coordinate (3.2, 0). The value of h determines the horizontal position of the vertex, while k determines its vertical position. Knowing the location of the vertex helps us to easily sketch the graph and to understand the behavior of the quadratic function around its peak.

Y-intercept

The y-intercept of a graph is the point where the curve intersects the y-axis. For quadratic functions, this intercept provides valuable information about the function's graph when x is zero. To find the y-intercept in our example, we substitute x with 0 in the function y = 0.1(x - 3.2)^2. After calculation, we find that y equals 1.024 when x is 0, thus, the y-intercept is the point (0, 1.024).

This process of setting x to zero simplifies finding the y-intercept for any quadratic function, especially when given in vertex form. The y-intercept is particularly useful when graphing the function because it marks the point where the graph starts or crosses the y-axis. Moreover, it's a handy reference point when checking the symmetry of the graph concerning the vertex.

Vertex Form

The vertex form of a quadratic function is an algebraic expression that allows us to immediately identify the function's vertex. This form is particularly useful for graphing and analyzing the behavior of quadratics. It is expressed as y = a(x - h)^2 + k, where a is the coefficient that affects the width and direction of the parabola, and (h, k) is the vertex.

In a vertex form equation, any positive value of a means that the parabola opens upwards, and if a is negative, it opens downwards. The magnitude of a affects how 'stretched' or 'compressed' the parabola appears. As for the coordinates h and k, they reveal the exact location of the vertex on the graph, allowing for immediate graph sketching without the need for further calculations. The example y = 0.1(x - 3.2)^2 showcases a parabola that opens upwards with a vertex at (3.2, 0). By learning to work with the vertex form, students can efficiently graph quadratic functions and comprehend their key characteristics.