Problem 66 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the vertex. $$y=-x^{2}+4 x+16$$

Step-by-Step Solution

Answer

The graph of the function \(y = -x^2 + 4x + 16\) is a downward-opening parabola with its vertex at the point (2, 14).

Step 1: Identify the Type of the Function

The given function \(y = -x^2 + 4x + 16\) is a quadratic function in the form \(y = ax^2 + bx + c\), where \(a = -1\), \(b = 4\), and \(c = 16\). Since \(a = -1\), the parabola will open downwards.

Step 2: Calculate the Vertex

The vertex of the parabola is at \(x = -b/2a\). Plug \(b = 4\) and \(a = -1\) into the equation: \(x = -4 / (2 * -1) = 2\). Substitute \(x = 2\) into the function to find the y-coordinate: \(y = -(2)^2 + 4 * 2 + 16 = 14\). Therefore, the vertex is at (2, 14).

Step 3: Sketch the Graph

Plot the vertex at point (2, 14). Since the parabola opens downwards, sketch the graph with the branches heading downwards from the vertex.

Key Concepts

Vertex Form

The vertex form of a quadratic function is an essential way of expressing the equation that makes it easy to identify important features of its graph, particularly the vertex of the parabola. In general, a quadratic function can be written in vertex form as follows: \[ y = a(x-h)^2 + k \] where:

• \(a\) is a coefficient that affects the direction and width of the parabola.
• \((h, k)\) are the x and y coordinates of the vertex of the parabola.

This form highlights the vertex directly, making it easy to graph. You can convert a quadratic function from the standard form \(y = ax^2 + bx + c\) to vertex form by completing the square. Once you have the function in vertex form, graphing is more intuitive, and you can explore further transformations such as shifting and reflecting.

Parabola

A parabola is a U-shaped curve that represents the graph of a quadratic function. Quadratic functions are polynomial functions with a degree of two, and their general form can be expressed as \(y = ax^2 + bx + c\). The properties of a parabola are influenced by the coefficient \(a\):

• If \(a > 0\), the parabola opens upwards, creating a U-shaped curve.
• If \(a < 0\), the parabola opens downwards, forming an inverted U-shape.

The vertex of the parabola is a crucial point where it changes direction. For a quadratic function given in standard form, the vertex can be calculated using the formula:\[ x_{vertex} = -\frac{b}{2a} \]The y-coordinate can then be found by substituting this x-value back into the function. Understanding these characteristics can help in sketching accurate graphs of quadratic functions.

Graphing Quadratic Equations

Graphing quadratic equations involves several steps that help portray their structure visually. Here's a simplified guide:1. **Identify the Function**: Recognize that the given equation is quadratic, looking at its standard form \(y = ax^2 + bx + c\).2. **Calculate the Vertex**: Use the vertex formula \(x = -\frac{b}{2a}\) to find the x-coordinate. Substitute to find the y-coordinate, and you have the vertex point \((h, k)\).3. **Determine the Opening**: Check the sign of \(a\). If positive, the parabola opens upwards. If negative, it opens downwards.4. **Sketch the Graph**:

• Place the vertex on the coordinate plane.
• Draw the axis of symmetry, a vertical line through the vertex.
• Plot additional points by choosing x-values and calculating corresponding y-values for more accuracy.
• Reflect these points across the axis of symmetry for balance.
• Draw a smooth curve through all points.

Remember, practice improves understanding, and sketching these graphs frequently will make these steps easier and your sketches more accurate.