Problem 44 - 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. $$ y=x^{2} $$

Step-by-Step Solution

Answer

The graph of the given function opens upwards. The vertex of the graph is at the point (0,0). The equation of the axis of symmetry is \(x=0\).

Step 1: Determining the Opening Direction of the Function Graph

A quadratic function is of the form \(y=ax^{2}+bx+c\). The graph of the quadratic function opens up when the leading coefficient a is positive and opens down when a is negative. In this case, the quadratic function is \(y=x^{2}\). The leading coefficient a is 1, which is positive. Therefore, the graph opens upwards.

Step 2: Find the Vertex

The vertex of a quadratic function in the standard form \(y=ax^{2}+bx+c\) is given by the coordinates \((-b/2a , f(-b/2a))\). For this function \(y=x^{2}\), a=1, b=0, c=0. So, the x-coordinate of the vertex is \(-b/2a = -0/2*1 = 0\). To find the y-coordinate, replace x with 0 in the function to get \(y = (0)^{2} = 0\). So, the vertex is (0,0).

Step 3: Write the Equation of the Axis of Symmetry

The axis of symmetry of a quadratic function in the form \(y=ax^{2}+bx+c\) is given by the line \(x = -b/2a\). For this function, a=1 and b=0. So, the axis of symmetry is \(x = -0/2*1 = 0\). Therefore, the equation of the axis of symmetry is \(x = 0\).

Key Concepts

Graph of a Quadratic Function

When we talk about a quadratic function, we are referring to an equation that can be written in the standard form of \(y=ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The graph of this equation creates a U-shaped curve called a parabola.

The most distinctive feature of the parabola is whether it opens upwards or downwards, which depends solely on the sign of coefficient \(a\). If \(a\) is positive, as with \(y=x^2\), the parabola opens upward, resembling a cup that can hold water. Conversely, if \(a\) were negative, the graph would open downward, like an upside-down cup. The width of the parabola is also influenced by the value of \(a\); a smaller absolute value makes the parabola wider, while a larger absolute value makes it narrower.

If we plot \(y=x^2\) on a coordinate grid, every point on the graph follows this equation. Notice how symmetry plays an integral role in the shape of the graph, which we'll explore further in the section about the axis of symmetry.

Vertex of a Quadratic Function

The vertex of a quadratic function is the highest or lowest point on the graph of the parabola. For \(y=ax^2+bx+c\), we can find the vertex by using the formula \(x_v = -b/(2a)\), where \(x_v\) is the x-coordinate of the vertex. The y-coordinate is then found by substituting \(x_v\) back into the original equation.

For our example, we have \(y=x^2\), which means \(a=1\) and \(b=0\). Using our formula, we find the x-coordinate of the vertex to be \(-b/(2a) = -0/(2*1) = 0\). Substituting this back into the equation, the y-coordinate is \(y = (0)^2 = 0\). Therefore, the vertex of the function \(y=x^2\) is at the point (0,0).

The vertex is not only an important geometric feature, it also represents the function's maximum or minimum value, depending on whether the parabola opens down or up. In our case, since the parabola opens upwards, the vertex represents the minimum value.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that divides the parabola into two mirror images. This axis runs through the vertex, making the x-coordinate of the vertex particularly significant.

In the equation \(y=ax^2+bx+c\), the equation of the axis of symmetry can be found using the same formula for the x-coordinate of the vertex, \(x=-b/(2a)\). For the function \(y=x^2\), this gives us the axis of symmetry at \(x=0\), which means the line \(x=0\) is the axis; it's the y-axis itself in this case.

Knowing the axis of symmetry is incredibly useful for graphing and understanding quadratic functions because it informs us about the balance and distribution of the parabola's points. Whether we're sketching a parabola by hand or using a tool to generate its graph, the axis of symmetry helps ensure we have accurately positioned the key features of our quadratic function.