Question

graph the equation. $y = \frac{2}{3}x^2 + 4x + 4$

Explanation:

Step1: Identify the function type

The equation is \( y = \frac{2}{3}x^2 + 4x + 4 \), which is a quadratic function (since the highest power of \( x \) is 2). The general form of a quadratic function is \( y = ax^2 + bx + c \), here \( a=\frac{2}{3} \), \( b = 4 \), \( c = 4 \).

Step2: Find the vertex

For a quadratic function \( y=ax^2+bx+c \), the x - coordinate of the vertex is given by \( x=-\frac{b}{2a} \). Substitute \( a = \frac{2}{3} \) and \( b = 4 \) into the formula:
\( x=-\frac{4}{2\times\frac{2}{3}}=-\frac{4}{\frac{4}{3}}=- 3 \)
Then find the y - coordinate by substituting \( x=-3 \) into the function:
\( y=\frac{2}{3}(-3)^2+4(-3)+4=\frac{2}{3}\times9 - 12 + 4=6-12 + 4=-2 \)
So the vertex is at \( (-3,-2) \).

Step3: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function:
\( y=\frac{2}{3}(0)^2+4(0)+4 = 4 \). So the y - intercept is \( (0,4) \).

Step4: Find the axis of symmetry

The axis of symmetry of a quadratic function \( y = ax^2+bx + c \) is the vertical line \( x=-\frac{b}{2a} \), which we already found as \( x=-3 \).

Step5: Find another point

Let's choose \( x = - 6 \). Substitute \( x=-6 \) into the function:
\( y=\frac{2}{3}(-6)^2+4(-6)+4=\frac{2}{3}\times36-24 + 4=24-24 + 4 = 4 \). So the point \( (-6,4) \) is on the graph.

Step6: Plot the points and draw the parabola

Plot the vertex \( (-3,-2) \), the y - intercept \( (0,4) \), the point \( (-6,4) \) and use the axis of symmetry \( x = - 3 \) to draw the parabola. Since \( a=\frac{2}{3}>0 \), the parabola opens upwards.

Answer:

To graph \( y=\frac{2}{3}x^2 + 4x + 4 \):

1. The vertex is at \( (-3,-2) \), y - intercept at \( (0,4) \), another point at \( (-6,4) \), axis of symmetry \( x=-3 \).
2. Plot these points and draw a parabola opening upwards (since \( a=\frac{2}{3}>0 \)) passing through these points.