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The y-intercept of a quadratic function is the point where the graph crosses the y-axis. To find it, we set the value of x to zero in the equation. For the given quadratic function \(y = -x^{2} + 3x + 10\), substituting \(x = 0\) gives us:

\ y = -0^{2} + 3(0) + 10 = 10 \

Therefore, the y-intercept is (0, 10). This point is essential because it indicates where the parabola touches the y-axis. In graphical terms, this is the point at which the graph starts or ends, depending on the direction it opens.

X-intercepts are the points where the graph crosses the x-axis. To find them, we set the value of y to zero and solve for x. For the quadratic equation \(y = -x^{2} + 3x + 10\), setting \(y = 0\) leads us to:

\(0 = -x^{2} + 3x + 10\)
Rearrange it as: \(-x^{2} + 3x + 10 = 0\)
This is a standard form quadratic equation \(ax^{2} + bx + c = 0\). We can solve it using the quadratic formula:

\ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \

Substituting the values of \(a, b,\) and \(c\) into the formula gives:

\ x = \frac{-3 \pm \sqrt{3^{2} - 4(-1)(10)}}{2(-1)} \
Simplifying, we get:

\ x = \frac{-3 \pm \sqrt{9 + 40}}{-2} \
\ x = \frac{-3 \pm \sqrt{49}}{-2} \
\ x = \frac{-3 \pm 7}{-2} \
The two solutions are:

\( x = \frac{-3 + 7}{-2} = -2 \) and \( x = \frac{-3 - 7}{-2} = 5 \)

Therefore, the x-intercepts are (-2, 0) and (5, 0). These points are crucial for sketching the graph as they show where the parabola intersects the x-axis.

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^{2} + bx + c = 0\). It is given by:

\ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \

This formula derives from completing the square method and works for any quadratic equation, providing real or complex solutions.
To use it, you need:

• The coefficients \(a, b,\) and \(c\) from your quadratic equation\
• Substitute these values into the formula\
• Simplify under the square root and the rest of the components\
• Solve for x.\

In our example, \(y = -x^{2} + 3x + 10\), we had \(a = -1, b = 3,\) and \(c = 10\). Substituting these into the formula, we solved for x to find our x-intercepts. The quadratic formula proves essential in many areas of mathematics and is a must-know topic in algebra.

A parabola is the graph of a quadratic function and has a characteristic U-shape. For a standard quadratic equation \(y = ax^{2} + bx + c \), the direction the parabola opens depends on the sign of the coefficient \(a \).

• If \(a > 0\), the parabola opens upwards.\
• If \(a < 0\), the parabola opens downwards.\

In our equation \(y = -x^{2} + 3x + 10\), \(a = -1\), so the parabola opens downwards. Key features of a parabola include:

• Vertex: This is the highest or lowest point on the parabola. It can be found using \ x = \frac{-b}{2a} \ for its coordinates\
• Axis of Symmetry: A vertical line passing through the vertex, given by \ x = \frac{-b}{2a} \
• Intercepts: Points where the parabola crosses the x-axis and y-axis.\

Knowing these elements helps to sketch and understand the graph of any quadratic function more effectively.