Problem 13 - Precalculus Mathematics for Calculus

Question

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=x^{2}+4 x+3$$

Step-by-Step Solution

Answer

The vertex is \((-2, -1)\), \( x \)-intercepts are \( (-3, 0) \) and \( (-1, 0) \), and \( y \)-intercept is \((0, 3)\). Sketch the parabola based on these points.

Step 1: Express in Standard Form

The given quadratic function is already in standard form. The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \). Here, \( a = 1 \), \( b = 4 \), and \( c = 3 \). Thus, the function is expressed in standard form as \( f(x) = x^2 + 4x + 3 \).

Step 2: Find the Vertex

To find the vertex, we use the vertex formula \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 4 \), we get \( x = -\frac{4}{2(1)} = -2 \). Substitute \( x = -2 \) back into the function to find \( y \): \( f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1 \). So, the vertex is \((-2, -1)\).

Step 3: Find the x-intercepts

To find the \( x \)-intercepts, set \( f(x) = 0 \): \( x^2 + 4x + 3 = 0 \). Factor this quadratic to find \( (x + 3)(x + 1) = 0 \). Solving gives \( x = -3 \) and \( x = -1 \). Thus, the \( x \)-intercepts are \( (-3, 0) \) and \( (-1, 0) \).

Step 4: Find the y-intercept

To find the \( y \)-intercept, set \( x = 0 \): \( f(0) = 0^2 + 4(0) + 3 = 3 \). Thus, the \( y \)-intercept is \((0, 3)\).

Step 5: Sketch the Graph

The parabola opens upwards since \( a = 1 > 0 \). Plot the vertex at \((-2, -1)\), the \( x \)-intercepts at \((-3, 0)\) and \((-1, 0)\), and the \( y \)-intercept at \((0, 3)\). Connect these points to form the parabola. The axis of symmetry is the vertical line \( x = -2 \).

Key Concepts

Vertex Calculation

To find the vertex of a quadratic function, you'll want to focus on the formula for the vertex's x-coordinate: \( x = -\frac{b}{2a} \). This formula is vital because it provides the x-coordinate for the vertex of the parabola. In the function \( f(x) = x^2 + 4x + 3 \), we identify \( a = 1 \), \( b = 4 \), and \( c = 3 \).
Substituting these into the formula, we get:

• \( x = -\frac{4}{2 \times 1} = -2 \)

Once you have the x-coordinate, substitute it back into the original quadratic function to find the y-coordinate of the vertex:

• \( f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1 \)

Hence, the vertex of this quadratic function is \((-2, -1)\). This point represents the peak or the lowest point of the parabola, depending on whether it opens up or down.

X-intercepts

Finding the x-intercepts, or the roots, of a quadratic function involves determining where the function crosses the x-axis. For this, set the function equal to zero and solve for \( x \).
For \( f(x) = x^2 + 4x + 3 \), the equation becomes:

• \( x^2 + 4x + 3 = 0 \)

This needs to be factored into:

• \( (x + 3)(x + 1) = 0 \)

By solving this equation, we find the x-values:

• \( x = -3 \)
• \( x = -1 \)

Thus, the function intersects the x-axis at the points \((-3, 0)\) and \((-1, 0)\). These intercepts are crucial for understanding the spread of the parabola along the x-axis.

Y-intercept

The y-intercept of a quadratic function is the point where the graph crosses the y-axis. This is simply done by setting \( x \) to zero and calculating the value of the function.
For the function \( f(x) = x^2 + 4x + 3 \), the calculation is:

• \( f(0) = 0^2 + 4 \times 0 + 3 = 3 \)

Thus, the y-intercept is at the point \((0, 3)\). This point tells us where the parabola will cross the y-axis and is vital for plotting complete parabola graphs.

Graph Sketching

Graph sketching of a quadratic function involves plotting its various critical points and observing its shape. The primary points to consider are the vertex, x-intercepts, and y-intercepts.
For \( f(x) = x^2 + 4x + 3 \):

• Vertex: \((-2, -1)\) tells us the turning point and helps in drawing the symmetry of the graph.
• X-intercepts: \((-3, 0)\), \((-1, 0)\) show where the graph crosses the x-axis.
• Y-intercept: \((0, 3)\) reveals the crossing point with the y-axis.

With these points:

• Start by plotting the vertex as it helps in understanding the symmetry of the parabola.
• Next, plot the x-intercepts to define where the parabola cuts through the horizontal axis.
• Plot the y-intercept to mark where it crosses the vertical axis.

Finally, draw a symmetric curve through these points. Remember, since \( a = 1 \) which is positive, the parabola opens upwards. The function's axis of symmetry is the vertical line \( x = -2 \). This detailed sketch should help you visualize the quadratic function clearly.