Problem 27 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=-3 x^{2}+12 x-7$$

Step-by-Step Solution

Answer

The function is a quadratic, opening downwards with vertex at (2, 5) and points (1, 2) and (3, 2).

Step 1: Identify the Type of Function

The given function \( f(x) = -3x^2 + 12x - 7 \) is a quadratic function because the highest power of \( x \) is 2. Quadratic functions are typically of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

Step 2: Determine the Vertex

To find the vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \), use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 12 \), so the x-coordinate of the vertex is \( x = -\frac{12}{2(-3)} = 2 \). Substitute \( x = 2 \) into the function to find the y-coordinate: \( f(2) = -3(2)^2 + 12(2) - 7 = 5 \). Thus, the vertex is \( (2, 5) \).

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is the vertical line that passes through the vertex. Its equation is \( x = 2 \), where 2 is the x-coordinate of the vertex.

Step 4: Determine the Direction of the Parabola

Since \( a = -3 \) (a negative number), the parabola opens downwards. This means that as \( x \) moves away from the vertex, the function values decrease.

Step 5: Find Additional Points

To plot the graph more accurately, select additional x-values and compute their corresponding y-values. For instance: For \( x = 1 \): \( f(1) = -3(1)^2 + 12(1) - 7 = 2 \), yielding the point \( (1, 2) \). For \( x = 3 \): \( f(3) = -3(3)^2 + 12(3) - 7 = 2 \), yielding the point \( (3, 2) \). Note that these y-values confirm the symmetry around the vertex.

Step 6: Sketch the Graph

Start by plotting the vertex \( (2, 5) \) and the points \( (1, 2) \) and \( (3, 2) \). Draw the axis of symmetry \( x = 2 \). With these points, sketch the parabola opening downwards, ensuring it is symmetric around the axis.

Key Concepts

Vertex of a Parabola

In a quadratic function like \( f(x) = ax^2 + bx + c \), the vertex of the parabola plays a crucial role. This single point marks the peak or the valley of the graph, depending on its orientation. For a function opening downward, like \( f(x) = -3x^2 + 12x - 7 \), the vertex is the highest point, representing a maximum.

To find the vertex, we use the formula to determine the x-coordinate: \( x = -\frac{b}{2a} \). By substituting the coefficients, \( b = 12 \) and \( a = -3 \), we find \( x = 2 \).

Next, we substitute \( x = 2 \) back into the original function to uncover the y-coordinate. This calculation yields \( f(2) = 5 \). So, the vertex of this parabola is at \( (2, 5) \). It's important to remember that this point on the graph is where the function has its turning point.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For any quadratic function, this axis passes through the vertex, ensuring that the shape is evenly balanced on either side of it.

In our function \( f(x) = -3x^2 + 12x - 7 \), the axis of symmetry is easily identifiable using the x-coordinate of the vertex as \( x = 2 \). This axis can be viewed as a line \( x = 2 \), and all points on the parabola have a symmetric counterpart across this line.

In practical terms, the axis of symmetry helps when graphing the parabola, as any point with coordinates \((x, y)\) will have a corresponding point \((2-(x-2), y)\) along this axis.

Graphing Parabolas

Graphing a parabola involves plotting key points and understanding its shape. With the function \( f(x) = -3x^2 + 12x - 7 \), begin by marking the vertex \((2, 5)\) on your graph. This point acts as a reference for drawing the parabola.

Since the coefficient \( a = -3 \) is negative, the parabola opens downward, indicating that it forms a frown shape, with the vertex being the maximum point.

To enhance accuracy, plot additional points. Substitute values around the vertex, such as \( x = 1 \) and \( x = 3 \), obtaining points \((1, 2)\) and \((3, 2)\). Notice how they align symmetrically about the axis, which is \( x = 2 \).

Finally, connect these points, ensuring the curve remains smooth and reflects the natural symmetry. Graphing parabolas becomes a task of plotting, connecting, and recognizing patterns in symmetry and curvature.