Problem 13 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using the vertex, the \(y\) -intercept, and two other points, not including the \(x\) -intercepts. Check the graph using a calculator. \(y=2 x^{2}+3\)

Step-by-Step Solution

Answer

The parabola has a vertex at (0, 3), a y-intercept at (0, 3), and passes through (1, 5) and (-1, 5).

Step 1: Identify the Vertex

The given quadratic equation is in the form \(y = ax^2 + bx + c\), where \(a = 2\), \(b = 0\), and \(c = 3\). For a parabola in this form, the vertex can be identified using \(x = -\frac{b}{2a}\). For our equation, \(x = -\frac{0}{4} = 0\). Substitute \(x = 0\) back into the original equation to find \(y\): \(y = 2(0)^2 + 3 = 3\). Thus, the vertex is \((0, 3)\).

Step 2: Find the Y-Intercept

The y-intercept of a parabola is the point where the graph crosses the y-axis. This occurs when \(x = 0\). We have already calculated when \(x = 0, y = 3\). Therefore, the y-intercept is \((0, 3)\), which is also the vertex in this case.

Step 3: Identify Additional Points

To find additional points on the graph, select values for \(x\) and calculate the corresponding \(y\). For \(x = 1\), we calculate \(y = 2(1)^2 + 3 = 5\), giving the point \((1, 5)\). For \(x = -1\), calculate \(y = 2(-1)^2 + 3 = 5\), giving the point \((-1, 5)\).

Step 4: Graph the Points

Plot the vertex \((0, 3)\), the additional points \((1, 5)\) and \((-1, 5)\), and the y-intercept \((0, 3)\) on a coordinate plane. These points should form a symmetric parabola opening upwards because the coefficient of \(x^2\) is positive.

Step 5: Verify with a Calculator

Using a graphing calculator, input the equation \(y = 2x^2 + 3\) and observe the graph. Check that it matches the points and structure identified: vertex at \((0, 3)\), y-intercept at \((0, 3)\), and passing through \((1, 5)\) and \((-1, 5)\). The graph's symmetry and upward opening should align with the calculations.

Key Concepts

Understanding the Vertex

In any quadratic equation, the vertex is a key feature. It is the point where the parabola changes direction. For a quadratic equation in the standard form \(y = ax^2 + bx + c\), you can find the vertex using the formula for the x-value: \(x = -\frac{b}{2a}\). Once you have the x-value, substitute it back into the equation to find the y-coordinate. This gives you the vertex, represented as \((x, y)\).
For the equation \(y = 2x^2 + 3\), the vertex was calculated as \((0, 3)\). The vertex tells you a lot about the graph's shape and direction. Here, since the coefficient \(a = 2\) is positive, the parabola opens upwards. The vertex becomes the lowest point on the graph.

Determining the Y-Intercept

Finding the y-intercept of a parabola in a quadratic equation is straightforward. This is the point where the graph crosses the y-axis, and it occurs when \(x = 0\).
Substitute \(x = 0\) into your quadratic equation \(y = ax^2 + bx + c\). The value of \(y\) when \(x = 0\) is simply \(c\), the constant term. In this case, for \(y = 2x^2 + 3\), when \(x = 0\), \(y = 3\), so the y-intercept is \((0, 3)\).
Interestingly, the vertex and the y-intercept can sometimes be the same, as seen in this problem. This typically happens when the parabola is symmetric about the y-axis.

Exploring the Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. It has the general form \(y = ax^2 + bx + c\). The curve produced by this equation is called a parabola.
Key features of a parabola include its vertex, axis of symmetry, and its opening direction, which depends on the sign and value of the coefficient \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

The quadratic equation can describe real-world scenarios like projectile motion or optimizing areas, making it an important concept in algebra.

Using a Graphing Calculator

A graphing calculator is an excellent tool for visualizing quadratic equations. By inputting the equation, students can instantly see the parabola and verify calculated points such as the vertex and y-intercept.
Using a graphing calculator can enhance understanding:

• Confirm the vertex by checking if it's the point where the direction changes.
• See symmetry about the vertex, with points equidistant from it.
• Observe the intersection at the y-intercept point, making calculations tangible.

These calculators assist students in grasping abstract concepts by turning them into visual, interactive experiences.