Question

what information can i find from this form of the equation?
$f(x)=x^2 + 8x + 11$

1. vertex: $(8, 11)$
2. x - intercepts: $x = 8$ and $x = 11$
3. y - intercept: $y = 11$
4. y - intercept: $y = - 11$

Explanation:

Step1: Recall y - intercept definition

The y - intercept of a function \(y = f(x)\) is the value of \(y\) when \(x = 0\).

Step2: Substitute \(x = 0\) into \(f(x)=x^{2}+8x + 11\)

Substitute \(x = 0\) into the function: \(f(0)=0^{2}+8\times0 + 11\)
Calculate the right - hand side: \(f(0)=0 + 0+11=11\). So the y - intercept is \(y = 11\).

Step3: Analyze vertex

For a quadratic function in the form \(y=ax^{2}+bx + c\) (\(a
eq0\)), the x - coordinate of the vertex is given by \(x=-\frac{b}{2a}\). For \(f(x)=x^{2}+8x + 11\), \(a = 1\), \(b = 8\), so \(x=-\frac{8}{2\times1}=- 4\), and \(f(-4)=(-4)^{2}+8\times(-4)+11=16-32 + 11=-5\). So the vertex is \((-4,-5)\), not \((8,11)\).

Step4: Analyze x - intercepts

To find the x - intercepts, we set \(f(x)=0\), i.e., \(x^{2}+8x + 11 = 0\). Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 1\), \(b = 8\), \(c = 11\). The discriminant \(\Delta=b^{2}-4ac=64 - 44 = 20\), and \(x=\frac{-8\pm\sqrt{20}}{2}=\frac{-8\pm2\sqrt{5}}{2}=-4\pm\sqrt{5}\approx-4\pm2.24\), so the x - intercepts are not \(x = 8\) and \(x = 11\).

Answer:

The correct statement is "y - intercept: \(y = 11\)" (the orange card with this statement).