Question

Create a hand drawn sketch of the quadratic function: h(x)= -2x^2 +5x-8To earn full credit be sure to identify the vertex, the axis of symmetry and all intercepts. Include all work, calculations and steps needed (as described in this lesson) to create the graph.Let your teacher know if you have any questions on how to upload or share your written work.

Answer 1

Hello!

We have the function below:

`h\mleft(x\mright)=-2x^2+5x-8`

The first step is to identify the coefficients a, b and c:

• a = -2

,

• b = 5

,

• c = -8

Just by analyzing these coefficients, we can say that the concavity of the parabola will be facing downwards because the coefficient a is negative

1st step: using the quadratic formula, we will find the interceptions in the x-axis. Look:

`x=\frac{-b\pm\sqrt[]{b^2-4\cdot a\cdot c}}{2\cdot a}`

As we know the coefficients, let's replace them in the formula:

`\begin{gathered} x=\frac{-5\pm\sqrt[]{5^2-4\cdot(-2)\cdot(-8)}}{2\cdot(-2)} \\ \\ x=\frac{-5\pm\sqrt[]{25^{}-64}}{-4}=\frac{-5\pm\sqrt[]{-39}}{-4}= \end{gathered}`

Obs: as we obtained a negative square root, we can stop this step here. It means that

2nd step: let's identify the maximum point of it:

`\begin{gathered} x_V=(-b)/(2\cdot a)=(-5)/(2\cdot(-2))=(-5)/(-4)=1.25 \\ \\ y_V=-(b^2-4\cdot a\cdot c)/(4\cdot a)=-(5^2-4\cdot(-2)\cdot(-8))/(4\cdot(-2))=-(-39)/(-8)=-4.875 \end{gathered}`

So, the vertex is at the point (x, y) = (1.25, -4.875).

The axis of symmetry is x = 5/4 or also x = 1.25.

Look at the graph below:

Note: if you solve this function when x = 0, you will obtain one interception in the y-axis, look:

`\begin{gathered} h\mleft(x\mright)=-2x^2+5x-8 \\ h\mleft(0\mright)=-2\cdot0^2+5\cdot0-8 \\ h(0)=0+0-8 \\ h(0)=-8 \end{gathered}`

So, the y-intercept is at (x, y) = (0, -8) as you can see at the graph.