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Chapter 9: Problem 252

Find the intercepts of the parabola whose function is given. $$f(x)=4 x^{2}+4 x+1$$

Short Answer

Expert verified
Y-Intercept: (0, 1); X-Intercept: \left(-\frac{1}{2}, 0\right\)

Step by step solution

01

Identify the Y-Intercept

To find the Y-Intercept, set \(x = 0\) in the equation of the function and solve for \(f(x)\).\[f(0) = 4(0)^2 + 4(0) + 1 = 1\]So, the Y-Intercept is \((0, 1)\).
02

Find the X-Intercepts

To find the X-Intercepts, set \(f(x) = 0\) and solve for \(x\). This means solving the quadratic equation \[4x^2 + 4x + 1 = 0\]
03

Quadratic Formula

Use the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]for the equation \[4x^2 + 4x + 1 = 0\]Here, \(a = 4\), \(b = 4\), and \(c = 1\).
04

Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\).\[b^2 - 4ac = 4^2 - 4(4)(1) = 16 - 16 = 0\]
05

Solve for X-Intercepts

Since the discriminant is 0, there is exactly one real solution. Solve for \(x\) using the quadratic formula:\[x = \frac{-4 \pm \sqrt{0}}{8} = \frac{-4}{8} = -\frac{1}{2}\]So, the X-Intercept is \(\left(-\frac{1}{2}, 0\right)\).

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Key Concepts

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

Y-Intercept
The Y-Intercept of a function is the point where the graph crosses the Y-axis. This occurs when the value of x is 0. To find it in a given quadratic function like \(f(x) = 4x^2 + 4x + 1\), substitute x with 0 and solve for \(f(x)\).
As shown in the solution:
  • Set \(x = 0\)
  • \(f(0) = 4(0)^2 + 4(0) + 1 = 1\)
We can see that when x is 0, \(f(x)\) is 1. Thus, the Y-Intercept is (0, 1). This point tells you where the parabola touches the Y-axis.
X-Intercept
The X-Intercepts represent the points where the graph crosses the X-axis. In other words, they are the values of x where \(f(x) = 0\). For a given quadratic function, you set \(f(x) = 0\) and solve for x. Consider the example function \(f(x) = 4x^2 + 4x + 1\).
We get the equation:
\(4x^2 + 4x + 1 = 0\).
To find the X-Intercepts, we use the quadratic formula later explained in detail.
This process involves calculating the roots of the quadratic equation. For our example, it yields one X-Intercept because its Discriminant equals zero. The X-Intercept found here is \(\left(-\frac{1}{2}, 0\right)\) after solving the quadratic equation.
Quadratic Formula
The Quadratic Formula is an essential tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides the solutions (roots) of the equation and is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Where:
  • a, b, c are coefficients of the quadratic equation
  • The term under the square root, \(b^2 - 4ac\), is known as the Discriminant
Using this formula in our example function \(4x^2 + 4x + 1\), plug in \(a = 4\), \(b = 4\), and \(c = 1\):
\(x = \frac{-4 \pm \sqrt{4^2 - 4(4)(1)}}{8}\).
This step is crucial to finding the X-Intercepts.
Discriminant
The Discriminant is a key part of the quadratic formula, represented by \(b^2 - 4ac\). It determines the nature of the roots of the quadratic equation:
  • If the Discriminant is positive (\(>0\)), there are two distinct real roots.
  • If it is zero (\(=0\)), there is exactly one real root.
  • If it is negative (\(<0\)), there are no real roots but two complex roots.
In our example function \(4x^2 + 4x + 1\), we calculate:
\[b^2 - 4ac = 4^2 - 4(4)(1) = 16 - 16 = 0\].
Since the Discriminant is zero, our quadratic equation has exactly one real solution, leading to one X-Intercept \(\left(-\frac{1}{2}, 0\right)\). This helps you understand whether the parabola intersects the X-axis at one point, two points, or none at all.

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Most popular questions from this chapter

Solve each inequality algebraically and write any solution in interval notation. $$x^{2}-4 x+3>0$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=3 x^{2}-12 x+7$$

Find the intercepts of the parabola whose function is given. $$f(x)=4 x^{2}-20 x+25$$

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Graph each function using a vertical shift. $$h(x)=x^{2}-5$$
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