Question

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=x^{2}-4 x+3\)

Short answer

The roots of the equation are x = 1 and x = 3. The graph is symmetric with respect to the line x = 2. The vertex of the parabola is at (2, -1). When sketched, the graph is a U shaped curve with its vertex at bottom.

Step-by-step solution

Find the Roots (Intercepts)

We will start by finding the roots of the equation. The roots will be the x-intercepts of the graph. To find the roots, set \(y = 0\) and solve for \(x\). Therefore we will solve the equation \(0=x^{2}-4x+3\). This is a quadratic equation and can be solved in a variety of ways, including factoring, completing the square, or using the quadratic formula. In this case, factoring is the quickest method. The factored form is \(0=(x-3)(x-1)\), therefore \(x = 1\) and \(x = 3\) are roots of the equation and the x-intercepts of the graph.

Test for Symmetry

Next, we will check if the graph of the equation is symmetrical. A quadratic equation of the form \(y=a x^{2}+bx+c\) is symmetric with respect to the line \(x=-b/2a\). Here, \(a = 1\) and \(b = -4\). So, the line of symmetry is \(x=-(-4)/2*1=2\). As we can see, 1 and 3 are at an equal distance from 2. Therefore, the graph is symmetric about the line \(x=2\). This means that the vertex of the parabola, which is the highest or lowest point of the graph, is at \(x = 2\). We can substitute \(x = 2\) into our equation to get the y-coordinate of the vertex, which is \(y = 2^{2}-4*2+3 = -1\). So, the vertex is at (2, -1).

Sketch the Graph

With the x-intercepts (1,0) and (3,0), the vertex at (2,-1) and knowledge of the symmetry, we can now sketch the graph. Plot these points first: the vertex and the points where the graph intersects the x-axis. Thanks to the symmetry, we know the rest of the parabola will fall into place: it will be a U shape (since the coefficient of \(x^2\) is positive) with its vertex at the bottom. Draw a neat curve to complete the U shaped graph, and make sure it is as symmetric as possible.

Key concepts

Finding Roots of Quadratic Equation

Understanding how to find the roots of a quadratic equation is critical for graphing its corresponding parabola. The roots, or x-intercepts, are the points where the parabola crosses the x-axis. These are the solutions to the equation when set equal to zero, represented as y=0. Take the quadratic equation y = x^2 - 4x + 3. By setting y to zero and solving for x, we get an equation that can often be factored into a product of binomials.

In this case, we have 0 = (x - 3)(x - 1). The solutions to this equation are x = 3 and x = 1, which tell us where the graph intersects the x-axis. Visualizing these solutions on a graph provides valuable reference points for the parabola's shape. Moreover, these roots split the x-axis into intervals; the sign of the quadratic expression within each interval determines whether the parabola lies above or below the x-axis.

Symmetry in Quadratic Functions

Quadratic functions display a property of symmetry which greatly aids in graphing them. The graph of any quadratic function, a parabola, is always symmetric about a vertical line known as the axis of symmetry. This line passes through the vertex of the parabola, the point where the function reaches its maximum or minimum value.

For the function y = ax^2 + bx + c, the axis of symmetry can be found using the formula x = -b / (2a). Applying this to our equation y = x^2 - 4x + 3, with a=1 and b=-4, we find the axis of symmetry is x=2. This not only helps us locate the vertex but also ensures that each point on one side of the axis has a mirror point on the other side. When plotting the graph, this symmetry can be used to check the accuracy of the graph and ensures the parabola opens symmetrically.

Vertex of a Parabola

The vertex of a parabola is perhaps one of the most significant features to locate when graphing a quadratic equation. It's the peak point for a downward-opening parabola or the lowest point for an upward-opening one. The vertex provides the maximum or minimum value of the quadratic function, respectively.

For a quadratic equation of the form y = ax^2 + bx + c, the vertex can be found once you have the axis of symmetry. For our example, y = x^2 - 4x + 3, with the line of symmetry at x = 2, we substitute x in the equation to find the y-coordinate of the vertex. We get y = 2^2 - 4*2 + 3 = -1. Therefore, the vertex's coordinates are (2, -1). When graphing, start by plotting the vertex, and then use the roots and symmetry to complete the parabola. The vertex often serves as a turning point in the graph and is essential for understanding the function's overall shape and direction.