Problem 57 - Intermediate Algebra

Question

Use a graphing calculator to graph each equation. $$ x^{2}-2 x+y=6 $$

Step-by-Step Solution

Answer

Graph the parabola \( y = -x^2 + 2x + 6 \) with vertex at \((1, 7)\) and points \((0, 6), (2, 6)\).

Step 1: Rewrite the Equation

First, we need to rewrite the equation in a form that makes it easier to graph. The given equation is \( x^2 - 2x + y = 6 \). We can solve for \( y \) to put it into a function form: \[ y = -x^2 + 2x + 6 \].

Step 2: Identify the Type of Graph

The equation \( y = -x^2 + 2x + 6 \) is a quadratic equation in the form of \( y = ax^2 + bx + c \) where \( a = -1 \), \( b = 2 \), and \( c = 6 \). The negative coefficient of \( x^2 \) indicates that the graph is a downward-opening parabola.

Step 3: Find the Vertex of the Parabola

For a quadratic equation in the form \( y = ax^2 + bx + c \), the vertex \( (h, k) \) can be found using the formula: \( h = -\frac{b}{2a} \). Plugging in our values \( a = -1 \) and \( b = 2 \): \[ h = -\frac{2}{2(-1)} = 1 \] Substitute \( h = 1 \) back into the equation for \( y \): \[ k = -(1)^2 + 2(1) + 6 = 7 \] So, the vertex is \((1, 7)\).

Step 4: Determine the Axis of Symmetry

The axis of symmetry for a parabola \( y = ax^2 + bx + c \) is given by the line \( x = h \), where \( h \) is the x-coordinate of the vertex. For our equation, the axis of symmetry is \( x = 1 \).

Step 5: Choose Additional Points

To sketch the parabola, choose additional \( x \)-values around the vertex and substitute them into the equation to find corresponding \( y \)-values. This will help define the shape of the parabola. For example, choose \( x = 0 \) and \( x = 2 \): - When \( x = 0 \), \( y = -0^2 + 2(0) + 6 = 6 \) - When \( x = 2 \), \( y = -2^2 + 2(2) + 6 = 6 \)

Step 6: Graph the Parabola

Using the vertex, additional points, and the axis of symmetry, plot these points on the graph. Draw a smooth curve through the points to represent the downward-opening parabola centered at \( (1, 7) \), passing through \((0, 6)\) and \((2, 6)\), with axis of symmetry \( x = 1 \).

Key Concepts

Vertex of Parabola

The vertex of a parabola is a critical point that represents its highest or lowest point, depending on the parabola's orientation. For a quadratic function in the form of \( y = ax^2 + bx + c \), the vertex is located at \( (h, k) \), with formulas \( h = -\frac{b}{2a} \) and \( k \) being the functional value at this point. In our example, the equation \( y = -x^2 + 2x + 6 \) leads us to find the vertex at \( (1, 7) \).

• The value of \( a \) helps in determining the direction of the parabola. A negative \( a \) means it opens downwards, which is the case here.
• The vertex is often the maximum or minimum point. Since our parabola opens downwards, \( (1, 7) \) is a maximum point.
• Finding the vertex is crucial for sketching the parabola, as it guides the graph's symmetry.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through its vertex, dividing it into two mirror-image halves. For the function \( y = ax^2 + bx + c \), it is given by \( x = h \). The axis of symmetry makes graphing parabola simpler.

• In the example \( y = -x^2 + 2x + 6 \), the axis of symmetry is found to be \( x = 1 \).
• This line helps in easily determining the shape and coherence of the parabola. Points equidistant from the axis mirror each other in their \( y \)-values.
• Use the axis as a guideline when plotting additional points to sketch the parabola.

Quadratic Function

A quadratic function is described by the formula \( y = ax^2 + bx + c \). It models a curve known as a parabola, exhibiting a host of properties including symmetry and a vertex. Quadratics are fundamental in algebra and pre-calculus due to their appearances in various mathematical problems and real-world applications.

• The coefficient \( a \) decides the direction of the parabola (upwards if \( a > 0 \), downwards if \( a < 0 \)).
• It's essential to convert equations into the vertex form or standard quadratic form to graph them easily.
• Quadratic functions illustrate concepts such as projectile motion in physics, optimizing business solutions, and economy graphs.

Graphing Calculator

A graphing calculator is a valuable tool for visualizing complex mathematical equations including quadratic functions. It allows students to quickly plot graphs, find crucial points, and analyze mathematical properties effortlessly.

• Input the quadratic equation and the calculator displays its graph, helping in understanding its shape.
• Use graphing calculators to locate intercepts, vertex, and test various values rapidly.
• As an educational aid, it fosters deeper comprehension of graphing concepts beyond manual calculations.
• Utilize its technology for exploring different function transformations, such as shifting or compressing graphs.