Question

a. Write $y=2x^2+4x+5$in standard form.

b. use properties of standard form to graph the equation.

Short answer

a. The standard form for the given function is,$y=2{\left(x+1\right)}^2+3$.

b. The graph for the function $y=2x^2+4x+5$is,

Step-by-step solution

Step 1 Given equation is,

$y=2x^2+4x+5$.

Rewrite the function in $y=a{\left(x-h\right)}^2+k$form by completing the square.

$$\begin{gathered} y=2\left(x^2+2x\right)+5 \\ y=2\left(x^2+2x+1\right)+5-2 \\ y=2{\left(x+1\right)}^2+3 \end{gathered}$$

Part b Step src="data:image/svg+xml;base64,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" role="math" localid="1645283540696" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/59fc7a4f-99a7-41ce-aaa3-4d96a097e6f3.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220219%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220219T154939Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=ed75b1a52edbac2966aeb8cceff0200aa29f44a54485a0dbfbebe0bafd70c11c" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/59fc7a4f-99a7-41ce-aaa3-4d96a097e6f3.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220219%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220219T153549Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=d818a378b98d366eba4f26d0c63fefb31b4dcc32fd3b6b703687c7b0f2fe922a" 1 Identify the constants a,h,k.

$a=2,h=-1,k=3$.

Since $a=2$, the parabola opens upward.

The axis of symmetry is, $x=h$.

So, $x=-1$.

The vertex is, $\left(h,k\right)=\left(-1,3\right)$.

Find the $y-$intercept by substituting $x=0$in the given equation.

$$\begin{gathered} y=0+0+5 \\ y=5 \end{gathered}$$

The $y-$intercept is, $\left(0,5\right)$.

The point symmetric to $\left(0,5\right)$across the axis of symmetry is, $\left(-2,5\right)$.

Find the $x$-intercept by substituting $y=0$in the standard form of the given equation.

role="math" localid="1645284524867" $$\begin{gathered} 0=2{\left(x+1\right)}^2+3 \\ -3=2{\left(x+1\right)}^2 \\ {\left(x+1\right)}^2=\frac{-3}{2} \\ x+1=\pm \sqrt{\frac{-3}{2}} \end{gathered}$$

The square root of a negative number tells us the solutions are complex numbers. So there are no x-intercepts.

Step 2 Now plot the graph with the known values.