Problem 21 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Plot the given point and determine whether the point lies inside or outside the parabola. $$ \begin{aligned} &y=x^{2}-2 x+5\\\ &A(0,4) \end{aligned} $$

Step-by-Step Solution

Answer

The point A(0,4) lies inside the parabola \(y=x^{2}-2 x+5\).

Step 1: Sketch the function

The given function is in the form of a quadratic equation \(y=ax^{2}+bx+c\). Here, \(a=1, b=-2\) and \(c=5\). The graph is a parabola that opens upwards, since \(a > 0\). The formula for the vertex is \(h=-\frac{b}{2a}\) and \(k=f(h)\). Here, \(h = -\frac{-2}{2*1} = 1\) and \(k = (1)^2 - 2*1 + 5 = 4\). So the vertex of the parabola is (1, 4). The axis of symmetry is \(x=h=1\). Sketch the function with these details.

Step 2: Plot the point

Now, plot the given point \(A(0,4)\) in the same coordinate plane. Ascertain its position relative to the parabola.

Step 3: Determine the position of the point

If a point lies on the parabola, the x-coordinate of the point when substituted in the equation will yield the same y-coordinate. When substituting \(x=0\) in \(y=x^{2}-2 x+5\), we get \(y=5\). However, the y-coordinate of point A is 4, not 5, so point A does not lie on the parabola. As the parabola opens upwards, any point within the parabola will have a y-value less than the corresponding y-value on the parabola. Here, the y-value of point A is less than the respective y-value on the parabola, so it falls inside the parabola.

Key Concepts

Quadratic Equations

Quadratic equations are algebraic expressions that take the form of \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). The graph of a quadratic function is called a parabola. This shape is symmetrical and either opens upward or downward, depending on the sign of the coefficient \(a\).

When \(a > 0\), the parabola opens upwards, and when \(a < 0\), it opens downwards. The quadratic equation can also be factored, solved by completing the square, or using the quadratic formula \(x = \frac{{-b \pm \-sqrt{b^2-4ac}}}{{2a}}\) to find its roots. These roots represent the x-intercepts of the parabola on a graph.

Graphing Functions

Graphing functions involves plotting a curve or a line on the Cartesian plane according to an equation. For quadratic functions, the shape of the graph is a parabola. To graph a quadratic function:

First, identify the vertex, which is the highest or lowest point on the graph. Then, find the axis of symmetry, which is a vertical line that passes through the vertex and divides the parabola into two mirror images. Plot additional points by choosing x-values and calculating corresponding y-values, and then use these points to draw the parabola.

Improving Graphs

To improve the accuracy of the graph, calculate and plot several points on either side of the axis of symmetry. This ensures that the sketched parabola is well-defined.

Axis of Symmetry

The axis of symmetry in the context of parabolas is a vertical line that divides the graph into two identical halves. For the equation of the parabola given in the standard form \(y=ax^2+bx+c\), the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).

This axis is significant because it not only helps in graphing the parabola accurately but also indicates the x-coordinate of the vertex of the parabola. The vertex's y-coordinate is found by substituting the x-coordinate of the vertex into the original equation.

Vertex of a Parabola

The vertex of a parabola is the point where it changes direction, which corresponds to either the maximum or minimum value of the function. In the equation \(y=ax^2+bx+c\), if \(a > 0\), the vertex represents the minimum point; if \(a < 0\), it's the maximum point.

As mentioned, the x-coordinate of the vertex is obtained from the axis of symmetry formula \(x = -\frac{b}{2a}\), and the y-coordinate is found by calculating the value of \(y\) using this x-coordinate. The vertex provides valuable information when sketching parabolas and is a crucial feature to identify when graphing quadratic functions.