Problem 86 - Intermediate Algebra

Question

Find the coordinates of the vertex and the direction in which each parabola opens. A. \(y=-x^{2}+4 x+1\) B. \(x=-y^{2}+4 y+1\)

Step-by-Step Solution

Answer

A: Vertex (2, 5), opens downward. B: Vertex (5, 2), opens to the left.

Step 1: Identify Parabola Axis

For question A, identify that the parabola is in the form of a quadratic in terms of x. This suggests that the parabola opens either upward or downward. For question B, the parabola is in the form of a quadratic in terms of y, meaning it will open either to the left or right.

Step 2: Calculate Vertex for First Parabola (A)

The vertex form of a parabola is given by \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. For the quadratic equation \(y=-x^2+4x+1\), use the formula \(h = -\frac{b}{2a}\) to find the x-coordinate of the vertex. Here, \(a = -1\) and \(b = 4\), so \(h = -\frac{4}{2(-1)} = 2\). Substitute \(x = 2\) into the equation to find the y-coordinate: \(-2^2 + 4(2) + 1 = 4 + 1 = 5\). Hence, the vertex is \((2, 5)\).

Step 3: Determine Opening Direction for First Parabola (A)

In the equation \(y=-x^2+4x+1\), the coefficient \(a = -1\) is negative, which indicates that the parabola opens downward.

Step 4: Calculate Vertex for Second Parabola (B)

For the quadratic equation \(x=-y^2+4y+1\), the formula for the y-coordinate of the vertex is \(k = -\frac{b}{2a}\), where \(a = -1\) and \(b = 4\). So \(k = -\frac{4}{2(-1)} = 2\). Substitute \(y = 2\) into the equation to find the x-coordinate: \(-2^2 + 4(2) + 1 = 4 + 1 = 5\). Therefore, the vertex is \((5, 2)\).

Step 5: Determine Opening Direction for Second Parabola (B)

In the equation \(x=-y^2+4y+1\), the coefficient \(a = -1\) is negative, indicating that the parabola opens to the left.

Key Concepts

Vertex

In the study of parabolas, the vertex is a crucial point that represents either the maximum or minimum value of the parabola, depending on its opening direction.
It is basically the "turning point" of the curve. For a parabola given by the equation \(y = ax^2 + bx + c\),
the vertex \((h, k)\) can be found using the formula:

• The x-coordinate: \(h = -\frac{b}{2a}\)
• Substitute \(h\) back into the equation to find the y-coordinate \(k\)

For Example A: \(y = -x^2 + 4x + 1\)
Setting \(a = -1\) and \(b = 4\), we use \(h = -\frac{b}{2a} = 2\),
substituting \(2\) into the equation gives us \(k = 5\), making the vertex \((2, 5)\).
For Example B, where \(x = -y^2 + 4y + 1\),
the vertex can similarly be found by switching the roles of \(x\) and \(y\), giving it coordinates \((5, 2)\). Understanding this concept is key for graphing and analyzing parabolas.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, usually written in the form:\(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
This equation forms a parabola when graphed on the coordinate plane.
The solutions or "roots" of the quadratic equations found using methods like factoring, completing the square, or the quadratic formula:

• The Quadratic Formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)
• Provides solutions where the curve crosses the x-axis.

For parabolas such as in exercises A and B, the quadratic equations define the shape and position on the graph:
- For the equation \(y = -x^2 + 4x + 1\), this represents a parabola opening either upwards or downwards.
- Whereas \(x = -y^2 + 4y + 1\) describes one opening to the sides. Each equation requires separate analysis to understand its properties.

Opening Direction

The opening direction of a parabola refers to the orientation of its curve on a graph.
This is determined by the coefficient of the squared term of its quadratic equation.

• For a standard quadratic equation \(ax^2 + bx + c\), when \(a > 0\), the parabola opens upward.
• Conversely, when \(a < 0\), the parabola opens downward.

For Example A, \(y = -x^2 + 4x + 1\), the negative \(a = -1\) indicates a downward opening.
In Example B, the equation \(x = -y^2 + 4y + 1\), involves \(y\) instead of \(x\) as the squared variable:

• If \(a < 0\), the parabola opens to the left.
• If \(a > 0\), it opens to the right.

Thus, for \(a = -1\), Example B opens to the left, illustrating how coefficient signs dictate the curve's orientation in relation to the coordinate axes.