Problem 27 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and graph the function. $$ f(x)=-2 x^{2}+12 x $$

Step-by-Step Solution

Answer

Vertex is at (3, 18), opens downward; x-intercepts at (0, 0) and (6, 0).

Step 1: Determine the Parabola's Direction

The quadratic function is given as \( f(x) = -2x^2 + 12x \). The coefficient of \( x^2 \) is \(-2\), which is less than zero, indicating that the parabola opens downward.

Step 2: Rewrite the Function in Vertex Form

The standard form of a quadratic function is \( ax^2 + bx + c \). First, complete the square to rewrite the function in vertex form \( a(x-h)^2 + k \). Start by factoring out \(-2\) from the first two terms: \[ f(x) = -2(x^2 - 6x) \]

Step 3: Complete the Square

To complete the square inside the parentheses, take half of the coefficient of \( x \), square it, and add and subtract it inside the square:\( -2(x^2 - 6x + 9 - 9) \)= \(-2((x - 3)^2-9) \)

Step 4: Simplify the Expression

Distribute the \(-2\) and simplify:\[ f(x) = -2(x - 3)^2 + 18 \]Now the function is in vertex form \( a(x-h)^2 + k \), where the vertex \( (h, k) \) is \((3, 18)\).

Step 5: Find x-Intercepts

Set \( f(x) = 0 \) to find the x-intercepts:\[ 0 = -2(x-3)^2 + 18 \]Rearrange and solve for \( x \):\[ 2(x - 3)^2 = 18 \] \[ (x - 3)^2 = 9 \] \[ x - 3 = \pm\sqrt{9} \]\[ x = 3 \pm 3 \]The solutions are \( x = 6 \) and \( x = 0 \), so the x-intercepts are at (0,0) and (6,0).

Step 6: Find the y-Intercept

Substitute \( x = 0 \) into the original function to find the y-intercept:\[ f(0) = -2(0)^2 + 12(0) = 0 \]Thus, the y-intercept is (0, 0).

Step 7: Graph the Function

With the vertex at (3,18), x-intercepts at (0,0) and (6,0), and the parabola opening downward, plot these points on the graph. Sketch the parabola, noting its downward opening and the symmetry about the line \( x = 3 \).

Key Concepts

Vertex Form of a Quadratic Function

Understanding the vertex form of a quadratic function is essential for easily determining crucial properties of a parabola, such as the vertex, direction, and intercepts. The vertex form of a quadratic equation is given as \( f(x) = a(x-h)^2 + k \). In this expression:

• \( a \) indicates the degree of vertical stretch or compression and the direction of the parabola.
• The \( (h, k) \) provides the coordinates for the vertex of the parabola.

For the function \( f(x) = -2(x-3)^2 + 18 \), the vertex form directly gives us the vertex at (3, 18). In practice, converting a function from standard form \( ax^2 + bx + c \) to vertex form often involves a process called completing the square. This allows us to easily locate the vertex by identifying \( h \) and \( k \) in the expression.

Parabola Direction

The direction in which a parabola opens depends on the coefficient \( a \) in the quadratic function. When the quadratic equation is in the form \( f(x) = ax^2 + bx + c \), the sign of \( a \) determines the parabola's direction:

• If \( a > 0 \), the parabola opens upward, resembling a U-shape.
• If \( a < 0 \), the parabola opens downward, similar to an upside-down U.

In our example function \( f(x) = -2x^2 + 12x \), the coefficient \( a = -2 \), which means the parabola opens downwards. This tells us that the vertex is a maximum point on the graph. The visual representation of the parabola will show closing arms directing downwards, indicating decreasing values of \( f(x) \) as we move away from the vertex.

Finding X-Intercepts

The x-intercepts are the points where the graph of the function crosses the x-axis. These points occur when \( f(x) = 0 \). To find the x-intercepts, solve the equation for \( x \).In our quadratic function, first, we express it as the vertex form, \( f(x) = -2(x - 3)^2 + 18 \), and set it to zero to find the intercepts:\[ 0 = -2(x-3)^2 + 18 \]Solving this, we find two values for \( x \):

• \( x = 0 \)
• \( x = 6 \)

Thus, our x-intercepts are at points (0,0) and (6,0). These intercepts indicate where the graph of the function touches or crosses the x-axis, providing two points through which the parabola passes.

Determining Y-Intercepts

The y-intercept of a quadratic function is where the graph intersects the y-axis, which occurs when \( x = 0 \). This point represents the value of the function when no horizontal distance from the origin is considered.To find the y-intercept for the function \( f(x) = -2x^2 + 12x \), substitute \( x = 0 \):\[ f(0) = -2(0)^2 + 12(0) \]This calculation simplifies to 0, so the y-intercept is at (0, 0). Interestingly, for this particular function, the y-intercept coincides with one of the x-intercepts. Intercepts are valuable because they provide real-number solutions that help plot significant points on the graph, aiding in the construction and understanding of the parabola's shape.