Problem 69 - A Graphical Approach to College Algebra

Question

Sketch a graph of a quadratic function that satisfies each set of given conditions. Use symmetry to label another point on your graph. Vertex \((-2,-3) ;\) through \((1,4)\)

Step-by-Step Solution

Answer

The parabola is described by \( y = \frac{7}{9}(x + 2)^2 - 3 \) with symmetric points at \((1, 4)\) and \((-5, 4)\).

Step 1: Identify the Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. In this problem, the vertex is \((-2, -3)\), so our equation becomes \( y = a(x + 2)^2 - 3 \).

Step 2: Use the Given Point to Find the Coefficient 'a'

The parabola also passes through the point \((1, 4)\). Substitute \(x = 1\) and \(y = 4\) into the equation to find \(a\):\[ 4 = a(1 + 2)^2 - 3 \]\[ 4 = a \cdot 9 - 3 \]\[ 4 + 3 = 9a \]\[ 7 = 9a \]\[ a = \frac{7}{9} \].

Step 3: Write the Final Equation of the Quadratic Function

Now that we've found \(a\), substitute it back into the vertex form:\[ y = \frac{7}{9}(x + 2)^2 - 3 \].

Step 4: Determine Symmetric Point Using Axis of Symmetry

The axis of symmetry for the parabola is \(x = -2\). Since we have the point \((1, 4)\) on one side of the symmetry line, we find a point directly opposite \((1, 4)\) by using the symmetry. From \(-2\) to \(1\) is a distance of \(3\), so we go \(3\) units left of \(-2\) to find another point at \((-5, 4)\).

Step 5: Sketch the Parabola

Plot the vertex at \((-2, -3)\) and the symmetric points \((1, 4)\) and \((-5, 4)\) on a graph. Draw a smooth curve through these points to complete the sketch of the parabola.

Key Concepts

Vertex Form

A quadratic function can often be written in the vertex form, making it easier to identify key features of its graph. The expression for vertex form is:

• \( y = a(x - h)^2 + k \)

In this equation, \((h, k)\) represents the vertex of the parabola, which is the point where the curve changes direction. The letter \(a\) reflects how wide or narrow the parabola opens and whether it opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

In the provided problem, we are given the vertex at \((-2, -3)\). This immediately tells us that the parabola is centered around \(x = -2\) and its lowest point is at \(-3\). To fully determine the equation, we need another point. Here, the point \((1, 4)\) is on the parabola.

By substituting this point into the vertex equation, we solve for \(a\), which allows us to describe the entire function:

• \(a = \frac{7}{9}\)
• The final quadratic function: \(y = \frac{7}{9}(x + 2)^2 - 3\)

Symmetry in Graphs

Symmetry in graphs of quadratic functions is a powerful property, especially for parabolas. If a parabola has a vertex at \((h, k)\), the vertical line \(x = h\) acts as a mirror line.

• This is known as the "axis of symmetry".

On each side of this line, the parabola's shape and points will be mirror images.

Let's take our example with the vertex at \((-2, -3)\), meaning the axis of symmetry is the line \(x = -2\). If the point \((1, 4)\) is given on the right side of this axis, we can find the symmetric point on the left side. The logical process is pretty straightforward:

• Measure the horizontal distance from \(-2\) to the point's \(x\)-value: \(1\).
• This distance is \(3\) units.
• Use this distance to locate the symmetric point on the other side of \(x = -2\): \((-2 - 3 = -5)\).
• Thus, the symmetric point is \((-5, 4)\).

Parabola Axis of Symmetry

The axis of symmetry is more than just a line dividing the parabola evenly; it plays an essential role in understanding the behavior of quadratic functions. This line runs vertically through the vertex of the parabola and dictates that for every point \((x, y)\) on one side of the parabola, there is a corresponding point \((-x, y)\) on the opposite side.

• The equation for this line in the parabola with vertex form \((h, k)\) is simply \(x = h\).

In our context, for the parabola with vertex \((-2, -3)\), the axis of symmetry is \(x = -2\). This means any point on the graph is mirrored across this line.

Identifying and utilizing the axis of symmetry can greatly aid in graphing and understanding quadratic functions. In solving quadratic problems, the axis helps not only locate symmetric points but also reinforces the consistency in the parabola's structure. Moreover, recognizing this aspect simplifies many other calculations related to motion, physics, and optimization problems. This insight into symmetry makes graphing not only a matter of plotting points but an organized display of a predictable pattern.