Problem 26 - A Graphical Approach to College Algebra

Question

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=x^{2}-2 x+3$$

Step-by-Step Solution

Answer

The vertex of the function is (1, 2), and it graphs as an upward-opening parabola with axis of symmetry x = 1.

Step 1: Identify the Quadratic Function Coefficients

The quadratic function given is \( P(x) = x^2 - 2x + 3 \). Here we identify the coefficients as follows: \( a = 1 \), \( b = -2 \), and \( c = 3 \). These coefficients will be used in calculating the vertex of the parabola.

Step 2: Use the Vertex Formula

The formula to find the vertex \((h, k)\) of a quadratic function \( ax^2 + bx + c \) is \( h = -\frac{b}{2a} \). Substituting the identified coefficients, we calculate:\[h = -\frac{-2}{2 \times 1} = 1\]Now, substitute \( h = 1 \) back into the quadratic function \( P(x) \) to find \( k \):\[k = 1^2 - 2 \times 1 + 3 = 1 - 2 + 3 = 2\]Therefore, the vertex of the quadratic function is \((1, 2)\).

Step 3: Determine Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line and is given by \( x = h \). Since we found \( h = 1 \) from our vertex calculation, the axis of symmetry is \( x = 1 \). This line will help us sketch the graph accurately.

Step 4: Plot the Vertex and Graph the Parabola

Plot the vertex point \((1, 2)\) on a coordinate plane. Since the coefficient \( a = 1 \) is positive, we know the parabola opens upwards. Draw the parabola starting at the vertex, ensuring it is symmetric about the axis \( x=1 \). Include additional points for accuracy, such as trying \( x = 0 \), which yields \( P(0) = 3 \), and plotting \((0, 3)\). This helps in drawing the correct shape.

Key Concepts

Vertex Formula

Understanding the vertex formula is crucial in graphing quadratic functions. The vertex of a parabola can be found using the formula \( h = -\frac{b}{2a} \) from the standard quadratic equation \( ax^2 + bx + c \). This formula helps locate the horizontal position of the vertex. Once you have \( h \), you find \( k \), the vertical position, by substituting back into the quadratic equation.Let's consider the exercise given: \( P(x) = x^2 - 2x + 3 \). Here, the coefficients are \( a = 1 \), \( b = -2 \), and \( c = 3 \). By applying the vertex formula, we calculate:

• \( h = -\frac{-2}{2 \times 1} = 1 \)

Substituting \( h = 1 \) back into the equation gives us the \( k \) value:

• \( k = 1^2 - 2 \times 1 + 3 = 2 \)

Thus, the vertex of the parabola is at \( (1, 2) \). The vertex is a key point on the graph, commonly the highest or lowest point depending on the direction the parabola opens.

Axis of Symmetry

The axis of symmetry in a quadratic function provides balance to the parabola. It's a vertical line that acts as a mirror for the graph, always passing through the vertex. Consequently, any parabola has the formula \( x = h \), where \( h \) is obtained from the vertex \( (h, k) \).For the equation \( P(x) = x^2 - 2x + 3 \) with vertex \( (1, 2) \), the axis of symmetry is

• \( x = 1 \)

The importance of the axis of symmetry lies in ensuring that the parabola is reflected evenly on both sides. When sketching your graph, knowing this vertical line helps you maintain precision. Any point on the left side of this line will have a corresponding point at the same distance on the right side.In conclusion, the axis of symmetry is an essential guide to drawing accurate graphs of quadratic functions.

Graphing Parabolas

Graphing a parabola involves a step-by-step approach, beginning with the vertex and using the axis of symmetry for symmetrical placement of points. First, plot the vertex \( (1, 2) \) on the graph. This point is essential as it dictates the general shape and direction of the parabola.Since the leading coefficient \( a = 1 \) is positive, we know:

• The parabola opens upwards.

To sketch the parabola accurately, choose additional points on either side of the axis of symmetry, such as \( x = 0 \) which provides the point \( (0, 3) \). You may also choose \( x = 2 \) to find additional points for plotting.With these points:

• \( P(0) = 3 \) gives point \( (0, 3) \)
• \( P(2) = 3 \) ensures symmetry with point \( (2, 3) \)

Ensure the graph is symmetric across \( x = 1 \). Finally, smoothly draw your parabola, ensuring it passes through all plotted points, and accurately reflects the orientation and width dictated by the function. This exercise highlights the effects of each part of the quadratic equation on the shape and position of the parabola.