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Chapter 9: Problem 247

Find the intercepts of the parabola whose function is given. $$f(x)=x^{2}+6 x+13$$

Short Answer

Expert verified
The y-intercept is (0, 13), and there are no real x-intercepts.

Step by step solution

01

Finding the y-intercept

To find the y-intercept, set x to 0 and solve for f(0). Plug in x = 0 into the function: \[ f(0) = 0^{2} + 6 \times 0 + 13 = 13 \]Thus, the y-intercept is at (0, 13).
02

Finding the x-intercepts

To find the x-intercepts, set f(x) to 0 and solve for x: \[ 0 = x^{2} + 6x + 13 \]This is a quadratic equation. To solve it, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]where \( a = 1 \), \( b = 6 \), and \( c = 13 \).
03

Calculate the discriminant

Compute the discriminant, \( \Delta \), which is under the square root in the quadratic formula: \[ \Delta = b^{2} - 4ac = 6^{2} - 4 \times 1 \times 13 = 36 - 52 = -16 \]Since the discriminant is negative, there are no real x-intercepts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
The quadratic formula is a vital tool for solving quadratic equations. A quadratic equation is any equation that can be written in the form \[ ax^{2} + bx + c = 0 \]. The quadratic formula allows you to find the solutions (or roots)
  • for any quadratic equation.
It is given by:
\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \].Use it when you have a quadratic equation in the standard form \( ax^{2} + bx + c = 0 \).
To use the formula effectively:
  • Identify the coefficients \( a \), \( b \), and \( c \).
  • Plug those values into the formula.
  • Calculate the discriminant \( \Delta = b^{2} - 4ac \).
  • Solve for \( x \) using the quadratic formula.
The quadratic formula can deal with any quadratic equation, even when it doesn't factor easily.Just follow the steps carefully!
Discriminant in Quadratics
The discriminant is an important component of the quadratic formula.It helps determine the nature of the roots of a quadratic equation.You can find it under the square root sign in the quadratic formula:
\[ \Delta = b^{2} - 4ac \]Here's how the discriminant impacts the solutions:
  • If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
  • If \( \Delta = 0 \), the quadratic equation has exactly one real root (a repeated root).
  • If \( \Delta < 0 \), the quadratic equation has no real roots, but instead has two complex roots.

In our exercise, we calculated the discriminant as follows:
  • \( b = 6 \)
  • \( a = 1 \)
  • \( c = 13 \)
  • \( \Delta = 6^{2} - 4 \times 1 \times 13 = 36 - 52 = -16 \)
A negative discriminant (-16 in our case) means there are no real x-intercepts but rather complex roots.
Finding Intercepts
Intercepts are points where a curve crosses the axes.In our quadratic function example, we looked for both the y-intercept and x-intercepts.
  1. Finding the y-intercept:Set \( x \) to 0 and solve for \( f(x) \).This will give you the point where the function crosses the y-axis.
    Example: For \( f(x) = x^{2} + 6x + 13 \), we have \( f(0) = 0^{2} + 6 \times 0 + 13 = 13 \).Thus, the y-intercept is at (0, 13).

  2. Finding the x-intercepts:Set \( f(x) \) to 0 and solve for \( x \).This gives the points where the function crosses the x-axis.Since this requires solving a quadratic equation, you often use the quadratic formula.
    Example: \( 0 = x^{2} + 6x + 13 \)Solving using the quadratic formula, we first find the discriminant and see if x-intercepts exist.In our case, \( \Delta = -16 \), which means there are no real x-intercepts (the function doesn't cross the x-axis).
Finding intercepts is crucial for understanding the shape and position of a graph on a coordinate plane.

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