Problem 2 - Algebra 2 and Trigonometry

Question

The graph of \(y=x^{2}-4 x+4\) is tangent to the \(x\) -axis at \(x=2\) and does not intersect the \(x\) -axis at any other point. How many roots does this function have? Explain your answer.

Step-by-Step Solution

Answer

The function has one root at \( x = 2 \).

Step 1: Identify the Given Function

The given function is a quadratic equation in the standard form: \[ y = x^2 - 4x + 4 \] which we will analyze using the properties of quadratic equations.

Step 2: Formulate the Discriminant

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by \( \Delta = b^2 - 4ac \). For our function, \( a = 1, b = -4, \) and \( c = 4 \) so the discriminant is: \[ \Delta = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0 \]

Step 3: Interpret the Discriminant Result

A discriminant of zero \( \Delta = 0 \) indicates that the quadratic function has exactly one distinct real root, which corresponds to the function being tangent to the \(x\)-axis and not intersecting it at any other point.

Step 4: Solve for the Root

Since the discriminant is zero, there is a repeated root at the vertex of the parabola. The vertex of the function \( y = x^2 - 4x + 4 \) can be found using the formula \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = -4 \): \[ x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2 \] This is the same \( x \) value given in the problem where the graph is tangent to the \( x \)-axis.

Step 5: Conclusion

The function has only one root at \( x = 2 \). This agrees with the information given, where the graph is tangent to the \(x\)-axis at one point and does not intersect it otherwise.

Key Concepts

Discriminant

The discriminant is a special value calculated from a quadratic equation that helps us understand the nature of the roots that the equation might have. In the context of the quadratic equation given, we calculated the discriminant using the formula: \[ \Delta = b^2 - 4ac \]For our specific equation, \[y = x^2 - 4x + 4,\] our coefficients are:

• \(a = 1\)
• \(b = -4\)
• \(c = 4\)

Substituting these into the formula gives us: \[\Delta = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0\]The discriminant (\(\Delta = 0\)) is particularly interesting because it tells us specific details about the roots. A discriminant of zero means that the equation has exactly one real root, also known as a double root or a repeated root. This is an important clue that helps define the graph's behavior.

Parabola

The parabola is the graphical representation of a quadratic function. In the quadratic equation \(y = x^2 - 4x + 4\), the shape we observe is a parabola. Every parabola can be characterized by some essential components:

• The vertex - the highest or lowest point of the parabola.
• The axis of symmetry - a vertical line that splits the parabola into two mirrored halves.
• The direction of opening - upwards or downwards, determined by the sign of \(a\).

In our equation, since \(a = 1\) and is positive, the parabola opens upwards. The vertex of the parabola can be calculated using the formula for the x-coordinate of the vertex: \[x = -\frac{b}{2a}\]Substituting the values, we find the vertex is at: \[x = -\frac{-4}{2 \cdot 1} = 2\]This vertex point is where the parabola is tangent to the x-axis (at \(x = 2\)), which means it just touches the x-axis, neither crossing it nor having another point of intersection.

Real Roots

A real root is a solution to a quadratic equation that is a real number. Using the discriminant, we can quickly determine the nature of the roots:

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root (a double root).
• If \(\Delta < 0\), the equation has no real roots.

For our equation \(y = x^2 - 4x + 4\), since the discriminant is zero, there is a single real root. We calculated this root: \[x = 2\]This single real root means the graph of the equation is only touching the x-axis at \(x = 2\), and it does not intersect at any other point.