Problem 71 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$G(x)=-6 x+x^{2}+5$$

Step-by-Step Solution

Answer

The graph of the given function G(x)=-6 x+x^{2}+5 is a upward facing parabola with its vertex at (3,-4) and intercepts at 1 and 5 on the x-axis. The axis of symmetry is the line x = 3.

Step 1: Rewriting the function in vertex form

To rewrite the given function \(G(x)=-6 x+x^{2}+5\) in vertex form, we first need to complete the square. We observe that the leading term in \(G(x)\) has a positive coefficient, so the graph opens upwards and the vertex will be the minimum point. The quadratic is of the form of \(ax^2 + bx + c\), hence we can rewrite this as \(G(x) = x^2 - 6x + 5\). If we complete the square, we get G(x) as \(G(x) = (x - 3)^2 - 4\). Therefore, the vertex of the function is (3, -4).

Step 2: Finding the axis of symmetry

The axis of symmetry of a quadratic function given in the vertex form \(y = a(x - h)^2 + k\) is the vertical line \(x = h\). Hence, for the given function, the axis of symmetry is the line \(x = 3\).

Step 3: Finding the x-intercepts

The x-intercepts are the points where the graph of the function intersects the x-axis. We find the x-intercepts by setting \(G(x) = 0\) and solving for \(x\). We thus have \(0 = (x - 3)^2 - 4\), which gives us \(x1 = 1\) and \(x2 = 5\)

Step 4: Sketching the graph

Now, use the vertex, axis of symmetry, and the x-intercepts obtained in the previous steps to sketch the graph. The vertex is the lowest point and is at (3, -4). The axis of symmetry divides the graph into two equal halves, and the x-intercepts are at 1 and 5. Therefore, the graph will be a downward facing parabola with its vertex at (3,-4) and intercepts at 1 and 5 on the x-axis.

Key Concepts

Vertex Form of a Quadratic

Understanding the vertex form of a quadratic function is crucial for quickly identifying its graph's most important features. A quadratic function can be expressed in vertex form as \( y = a(x-h)^2 + k \), where \( (h, k) \) are the coordinates of the vertex, and \( a \) determines the direction of the parabola (opening upwards for \( a > 0 \) and downwards for \( a < 0 \) ). The vertex represents the highest or lowest point on the graph, depending on whether the parabola opens up or down.

For the given function \( G(x) = -6x + x^2 + 5 \) when rewritten into the vertex form \( G(x) = (x - 3)^2 - 4 \) helps us see that the vertex is at \( (3, -4) \) without completing a whole square. This simple transformation allows students to easily find the most important point on the graph and understand the overall shape of the quadratic function's graph.

Axis of Symmetry

The axis of symmetry is a vertical line that bisects the parabola, and it passes through the vertex. In the vertex form \( y = a(x - h)^2 + k \), the axis of symmetry is the line \( x = h \).

For our equation \( G(x) = (x - 3)^2 - 4 \), the axis of symmetry would be \( x = 3 \). This means that every point on the left side of the line \( x = 3 \) has a mirror point on the right side with the same distance from the axis, creating symmetrical halves of the parabola.

X-Intercepts

The x-intercepts of a quadratic function are points where the graph crosses the x-axis. These occur where the function's value is zero. To find these intercepts from the vertex form \( y = a(x-h)^2 + k \), we set the equation to 0 and solve for \( x \).

In our example, \( G(x) = (x - 3)^2 - 4 \), we set it to zero and solve the resulting equation, leading us to \( x1 = 1 \) and \( x2 = 5 \). Therefore, the x-intercepts are at \( x = 1 \) and \( x = 5 \), providing points that guide us in sketching the graph accurately.

Completing the Square

Completing the square is a method used to convert a quadratic function in standard form, \( ax^2 + bx + c \), into vertex form. It involves creating a perfect square trinomial from the quadratic terms and adjusting the function accordingly.

To illustrate this with our function \( G(x) \), we take the original quadratic expression \( -6x + x^2 + 5 \) and arrange it as \( x^2 - 6x + 5 \), spotting the need to form a square term like \( (x - b/2)^2 \) and compensating by adding or subtracting constants. By doing this, we gained the vertex form and as a result, a clear path to identifying key features of the parabola.

Parabola Sketching

Creating a visual representation of a quadratic function, or sketching the parabola, involves marking the vertex, drawing the axis of symmetry, and plotting the x-intercepts. By doing so, the shape of the parabola becomes apparent, aiding in the comprehension of the function's behavior.

From our function \( G(x) = (x - 3)^2 - 4 \), we know the vertex is at \( (3, -4) \) and the axis of symmetry is \( x = 3 \). With x-intercepts at \( x = 1 \) and \( x = 5 \) marked, we can sketch a parabola opening downward. This leaves us with a complete picture showing the maximum or minimum point, points where the function crosses the x-axis, and the mirror-like symmetry of the graph.