Problem 47
Question
For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. \(\begin{array}{|c|c|c|c|c|c|}\hline x & {-2} & {-1} & {0} & {1} & {2} \\\ \hline y & {1} & {0} & {1} & {4} & {9} \\ \hline\end{array}\)
Step-by-Step Solution
Verified Answer
The quadratic function is \( y = x^2 + 2x + 1 \).
1Step 1: Identify the Vertex
From the table, examine the values of \(y\) that are generated by \(x\). The smallest \(y\) value (0) occurs when \(x = -1\), which suggests that the vertex is the point \((-1, 0)\).
2Step 2: Determine the Axis of Symmetry
The axis of symmetry for a quadratic function in vertex form \(y = a(x - h)^2 + k\) is \(x = h\). Since the vertex is \((-1, 0)\), the axis of symmetry is \(x = -1\).
3Step 3: Use Vertex Form to Write the Quadratic Function
With the vertex \((-1, 0)\), the function can be expressed in vertex form \(y = a(x + 1)^2 + 0\) or simply \(y = a(x + 1)^2\).
4Step 4: Determine the Value of \(a\)
Substitute another point from the table into the vertex form to find \(a\). Using point \((0, 1)\), substitute \((x, y) = (0, 1)\) into the equation: \(1 = a(0 + 1)^2\). This gives \(a = 1\).
5Step 5: Write the General Form of the Quadratic Function
Substituting \(a = 1\) back into the equation, the quadratic function is \(y = (x + 1)^2\). Expanding this form gives \(y = x^2 + 2x + 1\).
Key Concepts
Vertex FormAxis of SymmetryFinding the VertexGeneral Form of a Quadratic Equation
Vertex Form
The vertex form of a quadratic function is an important way to express the equation because it highlights the turning point of the parabola, known as the vertex. The vertex form is given by the expression \(y = a(x - h)^2 + k\), where \( (h, k) \) is the vertex of the parabola.
- **Vertex:** This is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. It provides crucial insight into the graph's shape and position.- **Parameter \(a\):** This determines the direction in which the parabola opens. If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards. It also affects the width of the parabola.- **Translation:** The values \(h\) and \(k\) indicate the vertex's horizontal and vertical translations from the origin.Understanding the vertex form is beneficial when you're analyzing the graph since it directly tells you the position and direction of the curve.
- **Vertex:** This is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. It provides crucial insight into the graph's shape and position.- **Parameter \(a\):** This determines the direction in which the parabola opens. If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards. It also affects the width of the parabola.- **Translation:** The values \(h\) and \(k\) indicate the vertex's horizontal and vertical translations from the origin.Understanding the vertex form is beneficial when you're analyzing the graph since it directly tells you the position and direction of the curve.
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic in vertex form \(y = a(x - h)^2 + k\), the axis of symmetry is always at \(x = h\). This line plays a critical role in graphing because:
- **Reflection:** Each point on the parabola that lies on one side of the axis of symmetry has a corresponding point directly opposite on the other side.
- **Vertex Location:** The axis of symmetry passes through the vertex, confirming its central role in the parabola's shape.- **Stability:** It ensures that the curve is balanced, supporting the symmetrical nature of quadratic functions.Finding the axis of symmetry is crucial when sketching or interpreting graphs as it gives you an immediate reference for the vertex location.
- **Reflection:** Each point on the parabola that lies on one side of the axis of symmetry has a corresponding point directly opposite on the other side.
- **Vertex Location:** The axis of symmetry passes through the vertex, confirming its central role in the parabola's shape.- **Stability:** It ensures that the curve is balanced, supporting the symmetrical nature of quadratic functions.Finding the axis of symmetry is crucial when sketching or interpreting graphs as it gives you an immediate reference for the vertex location.
Finding the Vertex
Finding the vertex is essential because it tells you the highest or lowest point on the graph, impacting decisions made based on the graph. When provided with a table of values, like in our exercise, identifying the vertex involves:
- **Analyzing Values:** Look for the point where the value of \(y\) is either minimized or maximized. In our example, this happens at \((-1, 0)\).- **Consistency:** Ensure that this point makes sense as a turning point by checking surrounding points.Once you have the vertex, which is \((h, k)\), you can apply this to the vertex form equation \(y = a(x - h)^2 + k\). This directly aids in both writing the equation and understanding the parabola's behavior.
- **Analyzing Values:** Look for the point where the value of \(y\) is either minimized or maximized. In our example, this happens at \((-1, 0)\).- **Consistency:** Ensure that this point makes sense as a turning point by checking surrounding points.Once you have the vertex, which is \((h, k)\), you can apply this to the vertex form equation \(y = a(x - h)^2 + k\). This directly aids in both writing the equation and understanding the parabola's behavior.
General Form of a Quadratic Equation
The general form of a quadratic equation is written as \(y = ax^2 + bx + c\). This form is highly useful for performing algebraic operations, such as solving quadratic equations or determining the function's roots. Here's why it's important:
- **Coefficient Insights:** The value of \(a\) determines the parabola's direction (upward or downward), while \(b\) and \(c\) affect the positioning.- **Finding Roots:** By applying the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), you can find where the function crosses the x-axis.- **Comparing Forms:** Converting from vertex form to general form shows the equation in a way that's often required in algebraic contexts. For example, expanding the vertex form \(y = (x + 1)^2\) produces \(y = x^2 + 2x + 1\), giving insight into how each coefficient affects the graph.Understanding these forms is crucial in navigating between different ways of looking at the quadratic function, enhancing both analytical and practical application skills.
- **Coefficient Insights:** The value of \(a\) determines the parabola's direction (upward or downward), while \(b\) and \(c\) affect the positioning.- **Finding Roots:** By applying the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), you can find where the function crosses the x-axis.- **Comparing Forms:** Converting from vertex form to general form shows the equation in a way that's often required in algebraic contexts. For example, expanding the vertex form \(y = (x + 1)^2\) produces \(y = x^2 + 2x + 1\), giving insight into how each coefficient affects the graph.Understanding these forms is crucial in navigating between different ways of looking at the quadratic function, enhancing both analytical and practical application skills.
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