Problem 72 - 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. Minimum value of \(-4\) at \(x=-3 ; y\) -intercept is \((0,3)\)

Step-by-Step Solution

Answer

The quadratic function is \( y = \frac{7}{9}(x+3)^2 - 4 \), with a symmetric point at \((-6, 3)\).

Step 1: Understanding 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 exercise, the minimum value of the quadratic function (which is the vertex) is given as \((-4, -3)\). This tells us that the vertex is \((-3, -4)\).

Step 2: Using the vertex information

We can substitute the vertex \((-3, -4)\) into the vertex form of the quadratic function, getting \( y = a(x+3)^2 - 4 \). We still need to find the value of \( a \).

Step 3: Plugging in the y-intercept

We know the function passes through \((0, 3)\), the y-intercept. Substitute these values into the vertex form equation: \( 3 = a(0+3)^2 - 4 \).

Step 4: Solving for the coefficient 'a'

Substituting in the y-intercept gives us \( 3 = 9a - 4 \). Solving for \( a \), we first add 4 to both sides to get \( 7 = 9a \), and then divide by 9 to get \( a = \frac{7}{9} \).

Step 5: Writing the complete quadratic equation

Substituting \( a = \frac{7}{9} \) into the vertex form of the equation, we get: \( y = \frac{7}{9}(x+3)^2 - 4 \).

Step 6: Identifying another point using symmetry

Using the axis of symmetry \(x = -3\), we can find a point symmetrical to \((0, 3)\). \( x = -3\) is the line of symmetry; since the distance from \(x = -3\) to \(x = 0\) is 3, another symmetrical point is \((-6, 3)\).

Key Concepts

Vertex Form

When it comes to quadratic functions, the vertex form is a powerful tool. This form is expressed as \( y = a(x-h)^2 + k \). Here, \((h, k)\) represents the vertex of the parabola, acting like a pinpointed location on the graph.
The "a" determines how open or narrow the parabola looks. If \(a\) is positive, the parabola opens upwards; if it's negative, it opens downwards.

• The vertex \((h, k)\) is a key point where the parabola changes direction. For our exercise, that point is \((-3, -4)\), meaning the lowest spot on our graph occurs at this coordinate.

Understanding the vertex form helps smoothly transition into other quadratic function characteristics, such as identifying the parabola’s curvature and direction.

Parabola

A parabola is the curved shape that's formed by graphing a quadratic function. Think of it like a gentle U or sometimes an upside-down U, depending on the direction it faces.
Each parabola includes:

• A vertex: the peak or the dip.
• An axis of symmetry: a vertical line passing through the vertex, dividing the shape into two identical parts.
• An opening direction determined by the sign of \(a\) in the vertex form equation.

Our specific exercise describes a parabola with a minimum point at \((-3, -4)\). This indicates it opens upwards, forming that familiar U shape that reaches its lowest at the vertex.

Y-intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. For our quadratic function, this point is given as \((0, 3)\).
It's easy to find because to do so, you just set \(x = 0\) in the function equation:
Substitute into \( y = \frac{7}{9}(x+3)^2 - 4 \):
\( y = \frac{7}{9}(0+3)^2 - 4 \) simplifies directly to \( y = 3 \), confirming the given y-intercept.

• The y-intercept plays a crucial role in sketching graphs, acting as a reference for symmetry and helping identify the shape's scale and correct placement.

Remember, the y-intercept is always present, even if it doesn't seem as crucial as the vertex for overall graph shape.

Symmetry in Graphs

Symmetry in graphs makes understanding and sketching parabolas easier. The axis of symmetry of a parabola with vertex form \( y = a(x-h)^2 + k \) is the vertical line \( x = h \).
This essentially splits the graph into two equal halves, so any point on one side has a twin over this axis.

• In our exercise, the axis of symmetry is \( x = -3 \).
• Given the y-intercept \((0, 3)\), you can find another point on the graph by reflecting across this symmetry line, resulting in \((-6, 3)\).

Symmetry helps to ensure that your graph is accurately drawn and that the parabola keeps its characteristic aesthetic on both sides.