Question

1. which of the following statements describe the key features of the graph of f(x) = -x² - 4x + 7? select all that apply. the vertex is (-2,19). the vertex is (-2,11). the function has a minimum value. the function has a maximum value. the y-intercept is (0,7).

Explanation:

Step1: Find the x - coordinate of the vertex

For a quadratic function \(f(x)=ax^{2}+bx + c\), the x - coordinate of the vertex is given by \(x =-\frac{b}{2a}\). For \(f(x)=-x^{2}-4x + 7\), \(a=-1\), \(b = - 4\). So \(x=-\frac{-4}{2\times(-1)}=-\frac{4}{2}=-2\).

Step2: Find the y - coordinate of the vertex

Substitute \(x = - 2\) into the function \(f(x)=-x^{2}-4x + 7\). \(f(-2)=-(-2)^{2}-4\times(-2)+7=-4 + 8+7=11\). So the vertex is \((-2,11)\).

Step3: Determine if it has a maximum or minimum

Since \(a=-1<0\), the parabola opens downwards, so the function has a maximum value (at the vertex).

Step4: Find the y - intercept

To find the y - intercept, set \(x = 0\). Then \(f(0)=-0^{2}-4\times0 + 7=7\). So the y - intercept is \((0,7)\).

Answer:

The vertex is \((-2,11)\), The function has a maximum value, The y - intercept is \((0,7)\) (corresponding to the options: The vertex is \((-2,11)\), The function has a maximum value, The y - intercept is \((0,7)\))