Problem 41 - Intermediate Algebra

Question

Write each equation of a parabola in standard form and graph it. Give the coordinates of the vertex. $$ y=-x^{2}-2 x+3 $$

Step-by-Step Solution

Answer

The vertex is at (1, 0) and the parabola opens downward.

Step 1: Identify the standard form of a parabola

The standard form of a parabola is given by the equation \( y = ax^2 + bx + c \). In this exercise, the given equation \( y = -x^2 - 2x + 3 \) is already in standard form where \( a = -1 \), \( b = -2 \), and \( c = 3 \).

Step 2: Determine the coordinates of the vertex

The vertex of the parabola in the standard form can be found using the vertex formula \( x = -\frac{b}{2a} \). Substituting the values, we have \( x = -\frac{-2}{2(-1)} = 1 \). Substitute \( x = 1 \) back into the original equation to find \( y \): \( y = -1^2 - 2(1) + 3 = 0 \). So, the vertex is at \((1, 0)\).

Step 3: Graph the parabola

Start by plotting the vertex \((1, 0)\). Since the coefficient \( a \) is negative, the parabola opens downwards. The y-intercept is \( c = 3 \), so another point is \((0, 3)\). Draw a symmetrical parabola downward through these points.

Key Concepts

Standard Form Equation of a Parabola

The standard form equation of a parabola is a foundational concept in algebra. It is presented in the form \( y = ax^2 + bx + c \). In this equation:

• \( a \), \( b \), and \( c \) are constants.
• \( a \) determines the direction of the parabola. If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards.
• \( b \) affects the horizontal placement of the parabola, while \( c \) is the y-intercept, indicating where the parabola crosses the y-axis.

For the equation \( y = -x^2 - 2x + 3 \), identifying that it is in standard form is straightforward since it matches the structure \( y = ax^2 + bx + c \) with \( a = -1 \), \( b = -2 \), and \( c = 3 \). This format is crucial because it helps unlock the properties needed to graph the parabola and find its vertex.

Vertex of a Parabola

The vertex is a key component in understanding the shape and position of a parabola. It serves as the turning point. For a parabola in standard form, \( y = ax^2 + bx + c \), the x-coordinate of the vertex is calculated using the formula \( x = -\frac{b}{2a} \).

• Substitute the values of \( b \) and \( a \) into this formula to find the x-coordinate.
• For the equation \( y = -x^2 - 2x + 3 \), substituting gives \( x = -\frac{-2}{2(-1)} = 1 \).
• Find the y-coordinate by substituting \( x = 1 \) back into the equation: \( y = -1^2 - 2(1) + 3 = 0 \).

Therefore, the vertex is located at \( (1, 0) \). This point not only tells us the maximum or minimum value of the parabola depending on its direction, but also helps in graphing the overall curve.

Graphing a Parabola

Graphing a parabola involves plotting its key points and understanding its geometric shape. Start by plotting the vertex of the parabola, which we found to be \( (1, 0) \). This is the critical point that dictates the symmetry of the parabola.

• Identify the direction of the parabola: since \( a = -1 \) is negative, the parabola opens downwards.
• Mark the y-intercept, which is given by the constant \( c = 3 \). This puts another point at \( (0, 3) \) on the graph.
• Use symmetry to find additional points across the axis of symmetry, which passes through the vertex. For example, if you have a point at \( (2, -3) \), there will be a mirrored point at \( (0, -3) \).

Finally, connect these points smoothly to form the parabolic curve. Make sure the graph is symmetrical around the vertex's axis, resulting in a U-shaped curve pointing downwards for this particular example.