Question

Hi Dear! Can you help me to solve this exercise,please!

Answer 1

k) (0, 5)

l) x = -2.5

m) (-2.5, -1.25)

n) (-1.382, 0) and (-3.618, 0)

Step-by-step explanation:

Part k

The y-intercept of f(x) is the value of f(x) when x = 0, so replacing it on the equation, we get:

f(x) = x² + 5x + 5

f(0) = 0² + 5(0) + 5

f(0) = 0 + 0 + 5

f(0) = 5

Then, the y-intercept of f(x) is the point (0, 5)

Part l

The line of symmetry for a quadratic function has the form x = -b/2a, where b is the number besides x and a is the number besides x². Replacing b = 5 and a = 1, we get:

x = -5/2(1)

x = -5/2

x = -2.5

Therefore, the line of symmetry is x = -2.5

Part m

The vertex is located at x = -2.5 because it is the line of symmetry. So, replacing x = -2.5 on the equation for f(x), we get:

f(x) = x² + 5x + 5

f(-2.5) = (-2.5)² + 5(-2.5) + 5

f(-2.5) = 6.25 - 12.5 + 5

f(-2.5) = -1.25

Then, the vertex of f(x) is the point (-2.5, -1.25)

Part n

The x-intercepts are the values of x that make f(x) = 0, so, we need to solve the following equation

f(x) = x² + 5x + 5 = 0

x² + 5x + 5 = 0

For this equation a = 1, b = 5, and c = 5, then we can use the quadratic equation to solve it as follows:

`\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-5\pm\sqrt[]{5^2-4(1)(5)}}{2(1)} \\ x=\frac{-5\pm\sqrt[]{5}_{}}{2} \end{gathered}`

So, the solutions are:

`\begin{gathered} x=\frac{-5+\sqrt[]{5}}{2}=-1.382 \\ \text{and} \\ x=\frac{-5-\sqrt[]{5}}{2}=-3.618 \end{gathered}`

Therefore, the x-intercepts are the points (-1.382, 0) and (-3.618, 0).

Part o)

Using the previous parts, we get that the graph is