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The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides a systematic way to find the roots of the equation, which are the values of \( x \) where the function equals zero. The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \( b^2 - 4ac \) is the discriminant, which helps in determining the nature and number of roots.
• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one root (also known as a repeated or double root).
• A negative discriminant indicates complex roots.

In our problem, for the function \( f(x) = 3x^2 - 13x - 10 \), the coefficients are \( a = 3 \), \( b = -13 \), and \( c = -10 \). By substituting these into the quadratic formula, we found the x-intercepts are \( x = 5 \) and \( x = -\frac{2}{3} \). This indicates where the graph crosses the x-axis.

Intercepts are crucial for understanding where a graph touches the axes. The **x-intercepts** are the points where the graph crosses the x-axis. To find them, we solve the quadratic equation by setting \( f(x) = 0 \). Using the quadratic formula, the x-intercepts for our function were found at \( x = 5 \) and \( x = -\frac{2}{3} \). These points tell us where the parabola intersects the x-axis, illustrating the function's real roots. The **y-intercept** is where the graph crosses the y-axis, found by substituting \( x = 0 \) in the quadratic function. For \( f(x) = 3x^2 - 13x - 10 \), the y-intercept was calculated as \( (0, -10) \). Understanding the intercepts helps in sketching the graph and provides insights into the function's behavior within its graph.

The graph of a quadratic function is a parabola. For the quadratic \( f(x) = 3x^2 - 13x - 10 \), the parabola opens upwards. This is determined by the sign of \( a \). Since \( a = 3 \) (positive), it implies the parabola opens upwards. A parabola has a symmetric curve, with the vertex being the highest or lowest point based on whether the parabola opens downward or upward.To sketch the graph:

• Locate the vertex; in this case, it's \( \left(\frac{13}{6}, -\frac{169}{12}\right) \).
• Plot the x-intercepts at \( (5, 0) \) and \( \left(-\frac{2}{3}, 0\right) \).
• Mark the y-intercept \( (0, -10) \).
• Draw the symmetrical parabola passing through these points, opening upwards.

The symmetric nature provides a balanced shape, which helps in predicting where points will fall based on one side of the vertex compared to the other.

The vertex of a parabola gives vital information about its graph. The vertex form of a quadratic function provides a clear pathway to sketch the parabola more easily:\[ f(x) = a(x-h)^2 + k \]Where \( (h, k) \) is the vertex and \( a \) decides the direction of the opening and the width of the parabola. For the function \( f(x) = 3x^2 - 13x - 10 \), the vertex \( (h, k) \) can be calculated using:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \)

By these calculations, the vertex was identified as \( \left(\frac{13}{6}, -\frac{169}{12}\right) \). This vertex form allows us to understand the transformation of the basic \( x^2 \) graph based on shifts and stretches/compressions. Not only does it pinpoint the peak or trough, but also helps in graphing the parabola accurately.