This series of problems will build your fluency with finding key features of quadratic functions algebraically. For each equation below, predict the number of distinct real roots then find the vertex and x-intercepts algebraically. ● An answer key is not provided for this task. Instead, it is expected that you confirm your answers using your graphing calculator and revise if necessary.

f(x) = 3x^2+ 12x – 21

1 answer

To analyze the quadratic function \( f(x) = 3x^2 + 12x - 21 \), we'll proceed step-by-step to find the number of distinct real roots, the vertex, and the x-intercepts.

Step 1: Predict the number of distinct real roots.

The number of distinct real roots of a quadratic function can be determined using the discriminant \( D \), calculated as follows:

\[ D = b^2 - 4ac \]

For the function \( f(x) = 3x^2 + 12x - 21 \):

  • \( a = 3 \)
  • \( b = 12 \)
  • \( c = -21 \)

Calculating the discriminant:

\[ D = 12^2 - 4(3)(-21) = 144 + 252 = 396 \]

Since \( D > 0 \), this indicates that there are two distinct real roots.

Step 2: Find the vertex.

The vertex \( (h, k) \) of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for \( h \):

\[ h = -\frac{b}{2a} \]

Substituting the values for \( a \) and \( b \):

\[ h = -\frac{12}{2 \times 3} = -\frac{12}{6} = -2 \]

Next, we find \( k \) by substituting \( h \) back into the function:

\[ k = f(-2) = 3(-2)^2 + 12(-2) - 21 \]

Calculating:

\[ k = 3(4) - 24 - 21 = 12 - 24 - 21 = -33 \]

Thus, the vertex of the parabola is at \( (-2, -33) \).

Step 3: Find the x-intercepts.

To find the x-intercepts, we set \( f(x) = 0 \) and solve for \( x \):

\[ 3x^2 + 12x - 21 = 0 \]

Dividing the entire equation by 3 gives:

\[ x^2 + 4x - 7 = 0 \]

Now using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \):

Here, \( a = 1 \), \( b = 4 \), and the discriminant \( D = 396 \).

Calculating the x-intercepts:

\[ x = \frac{-4 \pm \sqrt{396}}{2(1)} = \frac{-4 \pm \sqrt{396}}{2} \]

Simplifying \( \sqrt{396} \):

\[ \sqrt{396} = \sqrt{4 \times 99} = 2\sqrt{99} \]

Thus:

\[ x = \frac{-4 \pm 2\sqrt{99}}{2} = -2 \pm \sqrt{99} \]

Calculating the approximate values:

\[ \sqrt{99} \approx 9.95 \quad (\text{approximately}) \]

So the x-intercepts are:

\[ x \approx -2 + 9.95 \quad \text{and} \quad x \approx -2 - 9.95 \] \[ x \approx 7.95 \quad \text{and} \quad x \approx -11.95 \]

Summary of Results:

  • Number of distinct real roots: 2
  • Vertex: (-2, -33)
  • X-intercepts: \( x \approx 7.95 \) and \( x \approx -11.95 \)

Now you can confirm these results using a graphing calculator and revise them if necessary!

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