Problem 43 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=-10 x^{2}-7 x+2.66 $$

Step-by-Step Solution

Answer

The graph opens downwards. The vertex is at the point (0.35, 0.085). The equation of the axis of symmetry is \(x = 0.35\).

Step 1: Determine the direction of the Graph

In the given quadratic function, the coefficient 'a' is -10, which is less than zero. Therefore, the graph of the function opens downwards as indicated by negative value of the coefficient 'a'.

Step 2: Find the Vertex

The vertex of a quadratic function given in the standard form is at the point \((-b/2a, f(-b/2a))\). Here, 'a' is -10 and 'b' is -7. Thus, the x-coordinate of the vertex is \((-(-7)/2*(-10)) = 0.35\). Substituting this into the given function to find the corresponding y-coordinate, \(y = -10*(0.35)^2 - 7*0.35 + 2.66 = 0.085\). Therefore, the vertex is at the point (0.35, 0.085).

Step 3: Determine the Equation of the Axis of Symmetry

The equation of the axis of symmetry of a quadratic function given in standard form is \(x = -b/2a\). As found in the previous step, this is \(x = 0.35\). Therefore the equation of the axis of symmetry is \(x = 0.35\).

Key Concepts

Understanding the Axis of Symmetry

Quadratic functions are symmetrical, and their graphs are shaped like a parabola, which can either open upwards or downwards. The axis of symmetry is a vertical line that runs through the peak of the parabola, effectively splitting it into mirror images on either side.

This line is vital in understanding the behavior and attributes of the quadratic function. For the function given, the axis of symmetry is defined by the equation x = -b/2a. For our example, with a = -10 and b = -7, the axis of symmetry is x = 0.35. This line is not just a feature of the graph; it holds the key to finding the vertex, which is always located on the axis of symmetry.

Locating the Vertex of a Quadratic

The vertex of a quadratic function is the highest or lowest point on its graph, known as the parabola. It's a crucial element because it represents the maximum or minimum value of the function. You can find the vertex by using the coordinates \( (-b/2a, f(-b/2a)) \).

In our case, the function y = -10x^2 -7x + 2.66 has its vertex at the point \( (0.35, 0.085) \). To visualize this, imagine a parabola opening downwards (because the coefficient a is negative), it narrows to the point before it spreads out again — that point is the vertex. The x-value tells us where the vertex is horizontally, while the y-value determines its vertical position. The vertex gives you a significant advantage when graphing quadratics, as it serves as a reference point around which the parabola is shaped.

Graphing Quadratic Functions

When graphing quadratics, start by identifying the direction it opens, which depends on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if it's negative, as in the example y = -10x^2 -7x + 2.66, it opens downwards.

Next, locate the vertex which provides a central point to draw the parabola. Then plot a few more points by choosing x-values and calculating their respective y-values. Once you have these points, you can sketch the symmetry around the axis of symmetry from the vertex. Remember, the parabola is a smooth curve, not a series of connected lines. The axis of symmetry helps maintain the mirror-effect of the graph, ensuring accuracy and consistency. Graphing quadratics by hand can be iterative, but using the vertex and the axis of symmetry focuses the process, making it more efficient and precise.