Problem 5 - Basic Technical Mathematics with Calculus

Question

Sketch the graph of each parabola by using only the vertex and the \(y\) -intercept. Check the graph using a calculator. \(y=-3 x^{2}+10 x-4\)

Step-by-Step Solution

Answer

Vertex is \(\left(\frac{5}{3}, \frac{25}{9}\right)\), \(y\)-intercept is \((0, -4)\); the parabola opens downward.

Step 1: Identify the coefficients

The given quadratic equation is of the form \(y = ax^2 + bx + c\). For \(y = -3x^2 + 10x - 4\), identify \(a = -3\), \(b = 10\), and \(c = -4\).

Step 2: Find the vertex

The vertex of a parabola \(y = ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substituting in the values, we have \(x = -\frac{10}{2(-3)} = \frac{10}{6} = \frac{5}{3}\). Substitute \(x = \frac{5}{3}\) back into the equation to find \(y\): \(y = -3\left(\frac{5}{3}\right)^2 + 10\left(\frac{5}{3}\right) - 4\). Simplifying gives \(y = \frac{-75}{9} + \frac{50}{3} - 4\), which further simplifies to \(y = \frac{25}{9}\). Thus, the vertex is \(\left(\frac{5}{3}, \frac{25}{9}\right)\).

Step 3: Find the y-intercept

The \(y\)-intercept occurs where \(x = 0\). Substituting \(x = 0\) into the equation \(y = -3x^2 + 10x - 4\), we find \(y = -4\). Therefore, the \(y\)-intercept is at \((0, -4)\).

Step 4: Sketch the graph

Plot the vertex \(\left(\frac{5}{3}, \frac{25}{9}\right)\) and the \(y\)-intercept \((0, -4)\) on the coordinate plane. Since \(a = -3\) (which is negative), the parabola opens downward.

Step 5: Verify with a calculator

Use a graphing calculator to input the equation \(y = -3x^2 + 10x - 4\). Check that the vertex \(\left(\frac{5}{3}, \frac{25}{9}\right)\) and \(y\)-intercept \((0, -4)\) match the graph.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a pivotal point that helps us understand and sketch the graph more precisely. For any quadratic function expressed in the form \(y = ax^2 + bx + c\), the vertex provides the exact coordinates where the parabola either reaches a maximum or a minimum point—depending on its orientation. To find the vertex, we need the formula \(x = -\frac{b}{2a}\). Here, \(b\) and \(a\) are simply coefficients in the quadratic equation. Given the equation \(y = -3x^2 + 10x - 4\), plug in the coefficients \(b = 10\) and \(a = -3\) into the formula:

• \(x = -\frac{10}{2(-3)} = \frac{5}{3}\)

Once the x-coordinate of the vertex is found, substitute it back into the original equation to find the corresponding \(y\)-coordinate. This gives:

• \[y = -3\left(\frac{5}{3}\right)^2 + 10\left(\frac{5}{3}\right) - 4\]
• Simplifying yields \(y = \frac{25}{9}\)

So, the vertex of the parabola is \(\left(\frac{5}{3}, \frac{25}{9}\right)\). This key point helps to determine the parabola's highest point in the case of a downward-opening parabola.

Y-intercept Calculation

Understanding the \(y\)-intercept is crucial in graphing functions, as it reveals where the graph crosses the \(y\)-axis. The calculation of the \(y\)-intercept is straightforward. It's the point where the value of \(x\) is zero in the equation. By setting \(x = 0\) in the quadratic equation \(y = -3x^2 + 10x - 4\), we can solve for \(y\):

• Substituting gives \(y = -4\)

Thus, the \(y\)-intercept is at the point \((0, -4)\), providing clear insight into one fixed intersection point of the parabola with the \(y\)-axis. This point is especially useful when sketching the graph, as it gives a starting point along the vertical axis.

Downward-opening Parabola

A parabola’s orientation can be either upward or downward, determined by the coefficient \(a\) in the quadratic equation. When \(a\) is negative, as in \(-3\) in the equation \(y = -3x^2 + 10x - 4\), the parabola will open downwards. This means the vertex signifies a maximum rather than a minimum.This downwards-opening feature impacts the graph’s shape significantly:

• The arms of the parabola diverge downwards.
• The vertex serves as the highest point on the graph.
• The parabola forms a symmetrical shape around the vertical line drawn through the vertex.

Knowing these aspects helps one predict and understand the movement of the graph. Recognizing that the graph opens downwards translates into expecting the function's outputs to decrease as the value of \(x\) diverges from the vertex. Such understanding helps in real-life applications where maximizing certain outcomes is required.