Problem 41 - Calculus

Question

(a) If \(f(x)=a x^{2}+b x+c,\) what can you say about the values of \(a, b,\) and \(c\) if: (i) (1,1) is on the graph of \(f(x) ?\) (ii) (1,1) is the vertex of the graph of \(f(x) ?\) [Hint: The axis of symmetry is \(x=-b /(2 a) .]\) (iii) The \(y\) -intercept of the graph is (0,6)\(?\) (b) Find a quadratic function satisfying all three conditions.

Step-by-Step Solution

Answer

The function is \(f(x) = 5x^2 - 10x + 6\).

Step 1: Determine Condition (i)

To find out what can be said about \(a, b,\) and \(c\) if the point \((1, 1)\) is on the graph, we substitute \(x = 1\) and \(f(x) = 1\) into the equation.\[1 = a(1)^2 + b(1) + c.\]This simplifies to:\[a + b + c = 1\]This equation must hold true.

Step 2: Analyze Condition (ii)

The vertex is given as \((1, 1)\). For the vertex form of a quadratic function, the axis of symmetry is \(x = -\frac{b}{2a}\). Equating the x-coordinate of the vertex, we have:\[1 = -\frac{b}{2a} \]Solve for \(b\) in terms of \(a\):\[b = -2a.\]Thus, \(b\) must be \(-2a\) for the vertex to be \((1, 1)\).

Step 3: Examine Condition (iii)

For the y-intercept to be \((0, 6)\), we substitute \(x = 0\) into the function:\[f(0) = c = 6.\]So, the constant term \(c\) must be 6.

Step 4: Combine Conditions (i), (ii), and (iii)

Using our results from the previous steps, substitute \(b = -2a\) and \(c = 6\) into the equation from Step 1: \[a + (-2a) + 6 = 1.\]This simplifies to:\[-a + 6 = 1.\]Solving for \(a\), we find:\[a = 5.\]Therefore, \(a = 5, b = -10,\) and \(c = 6.\)

Step 5: Construct the Quadratic Function

Now that we have found \(a, b,\) and \(c\), we can construct the quadratic function:\[f(x) = 5x^2 - 10x + 6.\]This function satisfies all three conditions: 1. \((1, 1)\) is on the graph.2. \((1, 1)\) is the vertex.3. The y-intercept is \((0, 6)\).

Key Concepts

Vertex Form

The vertex form of a quadratic function provides a straightforward way to identify the vertex of the parabola. This form is given by \(f(x) = a(x-h)^2 + k\), where \( (h, k) \) is the vertex of the parabola.

Here, \(h\) determines the horizontal position of the vertex, while \(k\) affects the vertical position. The coefficient \(a\) influences the parabola's direction and whether it opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

By using the vertex form, you can easily determine the vertex by looking at the constants \(h\) and \(k\), without further calculation. This is extremely helpful in graphing the quadratic function and understanding its symmetry and shape.

In the context of the exercise, having the vertex at \((1, 1)\) indicates that in the vertex form \(f(x) = a(x-1)^2 + 1\), which confirms our vertex coordinates when considering the transformation properties of parabolas.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It divides the parabola into two mirror-image halves. For any quadratic function in the standard form \(f(x) = ax^2 + bx + c\), the formula for the axis of symmetry is given by \(x = -\frac{b}{2a}\).

This formula stems from completing the square, a method that rearranges the standard form into the vertex form. The axis of symmetry is particularly useful because it highlights the balance of the parabola around the vertex, allowing for easier graphing and comprehension of the parabola's properties.

In the exercise, with the vertex at \((1, 1)\), the axis of symmetry is \(x = 1\). From the formula \(x = -\frac{b}{2a}\) and substituting \(x = 1\), we derived that \(b = -2a\), which is crucial for setting up and validating the quadratic function through given conditions.

Y-intercept

The y-intercept is the point at which a graph intersects the y-axis. For any function \(f(x)\), this point occurs when \(x = 0\). Therefore, for a quadratic function \(f(x) = ax^2 + bx + c\), the y-intercept is simply the value \(c\), since \(f(0) = c\).

The y-intercept is an essential feature of any graph as it provides a starting point for plotting and gives insight into the graph's position relative to the y-axis.

In the provided exercise, setting the y-intercept to be \((0, 6)\) means that \(c = 6\). This information was used to solve for the remaining coefficients of the function, combined with other conditions like the vertex point. Knowing the y-intercept helps to quickly verify graph positions and understand the initial value of the function as it crosses the y-axis.