Example 9 Write a Quadratic Function Given Points Write a quadratic function for the graph that contains \( (-7,0),(-4,-30) \), and \( (1,0) \)
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To find the quadratic function given the points, we can use the fact that a quadratic can be expressed in the form \(y = a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the x-intercepts. Given the roots at \(x = -7\) and \(x = 1\), we can write: \[y = a(x + 7)(x - 1)\] We can substitute the point \((-4, -30)\) to find the value of \(a\). Plugging in: \(-30 = a(-4 + 7)(-4 - 1)\) results in \( -30 = a(3)(-5) \), therefore, \( -30 = -15a\), giving us \(a = 2\). The final function becomes: \[y = 2(x + 7)(x - 1)\] Which simplifies to: \[y = 2x^2 + 12x - 14\]. This equation will give you a beautiful parabola passing through the given points! Now, to ensure everything fits nicely, you can double-check by substituting back the points \((-7,0)\), \((-4,-30)\), and \((1,0)\) into your final equation. If they all satisfy the equation, congratulations, you've successfully derived the quadratic function!