Problem 37 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=-x^{2}+8 x+32 $$

Step-by-Step Solution

Answer

The graph of the given function opens downwards. The coordinates of the vertex are (4, 36). And the equation of the axis of symmetry is \(x = 4\).

Step 1: Determine the direction of the graph

Look at the coefficient of \(x^2\), which is \(a=-1\). Because \(a < 0\), the graph of the function opens downwards.

Step 2: Find the coordinates of the vertex

The x-coordinate of the vertex \(h\) is given by \(-b/2a\). Here \(a=-1\) and \(b=8\), so \(h = -(8)/(2*(-1)) = 4\). To find the y-coordinate of the vertex \(k\), substitute \(h=4\) into the equation to get \(k = -(4)^2 + 8*4 + 32 = 36\). So the coordinates of the vertex are (4, 36).

Step 3: Write the equation of the axis of symmetry

The equation of the axis of symmetry of a quadratic function is \(x = h\). Here, since \(h = 4\), the equation for the axis of symmetry is \(x = 4\).

Key Concepts

Axis of Symmetry

The axis of symmetry is a critical feature in understanding the structure of a parabola, which is the graph of any quadratic function. It is a vertical line that divides the parabola into two mirror images, ensuring that each side of the parabola is an exact reflection of the other.

This axis can be thought of as the fold line in a symmetrical figure; if you were to fold the parabola along this line, the two sides would align perfectly. Mathematically, the axis of symmetry for the quadratic function in the standard form of \( y = ax^2 + bx + c \) comes from the equation \( x = -\frac{b}{2a} \).

In the given exercise, where the quadratic function is \( y = -x^2 + 8x + 32 \), following Step 3 of the solution places the axis of symmetry at \( x = 4 \). This line is not only an important attribute for graphing but also for solving problems related to the function's symmetry, such as identifying the vertex or solving quadratic equations.

Vertex of a Parabola

The vertex of a parabola holds a position of remarkable prominence, as it represents the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. The vertex is where the function reaches its maximum or minimum value, which applies to real-world contexts like projectile motion or optimization problems.

For the quadratic function given in the exercise, \( y = -x^2 + 8x + 32 \), the process of finding the vertex begins with locating the x-coordinate, denoted as \( h \), by the formula (as per Step 2 of the solution) \( h = -\frac{b}{2a} \). The y-coordinate, denoted as \( k \), is then found by substituting \( h \) back into the original function.

Executing these steps, we determine the vertex of the function to be at the coordinates (4, 36). The coordinate (4, 36) doesn't solely help in sketching the graph; it also plays a crucial role in understanding the function's behavior and its extremum.

Graph of a Quadratic Function

The graph of a quadratic function is a parabola, which is a U-shaped curve that can open either upwards or downwards. Importantly, the direction of the opening is determined by the sign of the coefficient of the \( x^2 \) term in the quadratic function: If positive, the parabola opens upwards, if negative, it opens downwards.

Looking at our exercise's quadratic function, \( y = -x^2 + 8x + 32 \), the leading coefficient \( a \) is negative, indicating that the parabola opens downwards (as concluded in Step 1). This means the parabola's vertex is the highest point on the graph, representing a maximum.

The features such as the axis of symmetry and vertex help in drawing an accurate graph of the quadratic function. Other points can be plotted by choosing x-values and calculating their corresponding y-values to get a set of points that shape the curve. Therefore, knowing these fundamental components not only enhances comprehension but also simplifies the portrayal of quadratic functions visually.