Question

which equation choice could represent the graph shown below?

Explanation:

To determine the equation of the parabola, we analyze its key features:

Step 1: Identify the x - intercepts

The graph intersects the x - axis at \(x = 0\) and \(x=3\) (since it crosses the x - axis at the origin \((0,0)\) and at \((3,0)\)).

If a parabola has x - intercepts at \(x = r_1\) and \(x=r_2\), its factored form is \(y=a(x - r_1)(x - r_2)\). Here, \(r_1 = 0\) and \(r_2=3\), so the equation in factored form is \(y=a(x - 0)(x - 3)=ax(x - 3)=ax^{2}-3ax\)

Step 2: Determine the leading coefficient

The parabola opens upwards, so \(a>0\). Also, we can check the vertex or a point on the parabola. Let's assume a general form. If we take a simple case where \(a = 1\), the equation would be \(y=x(x - 3)=x^{2}-3x\)

Let's verify the vertex. The x - coordinate of the vertex of a parabola \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). For \(y=x^{2}-3x\), \(a = 1\) and \(b=- 3\), so \(x=-\frac{-3}{2\times1}=\frac{3}{2}=1.5\)

To find the y - coordinate, substitute \(x = 1.5\) into the equation: \(y=(1.5)^{2}-3\times(1.5)=2.25 - 4.5=- 2.25\)

Looking at the graph, the vertex seems to be around \((1.5,-2.25)\) or a similar point, which is consistent with the equation \(y=x^{2}-3x\) (or equivalent forms like \(y=(x)(x - 3)\))

Another way: If we consider the standard form \(y = ax^{2}+bx + c\). Since the parabola passes through \((0,0)\), when \(x = 0\), \(y = 0\), so \(c = 0\). The parabola also passes through \((3,0)\), so when \(x = 3\), \(y=0\). Let's assume the equation is \(y=x^{2}-3x\) (we can also have other equations with \(a>0\), but this is a simple one that fits the x - intercepts and the direction of opening)

If we expand \(y=x(x - 3)\), we get \(y=x^{2}-3x\)

So a possible equation for the graph is \(\boldsymbol{y=x^{2}-3x}\) (or equivalent equations like \(y = \frac{1}{2}x(x - 3)\) as long as \(a>0\))

(Note: If there were answer choices, we would check which of the choices has x - intercepts at \(x = 0\) and \(x = 3\) and \(a>0\))

Answer:

To determine the equation of the parabola, we analyze its key features:

Step 1: Identify the x - intercepts

The graph intersects the x - axis at \(x = 0\) and \(x=3\) (since it crosses the x - axis at the origin \((0,0)\) and at \((3,0)\)).

If a parabola has x - intercepts at \(x = r_1\) and \(x=r_2\), its factored form is \(y=a(x - r_1)(x - r_2)\). Here, \(r_1 = 0\) and \(r_2=3\), so the equation in factored form is \(y=a(x - 0)(x - 3)=ax(x - 3)=ax^{2}-3ax\)

Step 2: Determine the leading coefficient

The parabola opens upwards, so \(a>0\). Also, we can check the vertex or a point on the parabola. Let's assume a general form. If we take a simple case where \(a = 1\), the equation would be \(y=x(x - 3)=x^{2}-3x\)

Let's verify the vertex. The x - coordinate of the vertex of a parabola \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). For \(y=x^{2}-3x\), \(a = 1\) and \(b=- 3\), so \(x=-\frac{-3}{2\times1}=\frac{3}{2}=1.5\)

To find the y - coordinate, substitute \(x = 1.5\) into the equation: \(y=(1.5)^{2}-3\times(1.5)=2.25 - 4.5=- 2.25\)

Looking at the graph, the vertex seems to be around \((1.5,-2.25)\) or a similar point, which is consistent with the equation \(y=x^{2}-3x\) (or equivalent forms like \(y=(x)(x - 3)\))

Another way: If we consider the standard form \(y = ax^{2}+bx + c\). Since the parabola passes through \((0,0)\), when \(x = 0\), \(y = 0\), so \(c = 0\). The parabola also passes through \((3,0)\), so when \(x = 3\), \(y=0\). Let's assume the equation is \(y=x^{2}-3x\) (we can also have other equations with \(a>0\), but this is a simple one that fits the x - intercepts and the direction of opening)

If we expand \(y=x(x - 3)\), we get \(y=x^{2}-3x\)

So a possible equation for the graph is \(\boldsymbol{y=x^{2}-3x}\) (or equivalent equations like \(y = \frac{1}{2}x(x - 3)\) as long as \(a>0\))

(Note: If there were answer choices, we would check which of the choices has x - intercepts at \(x = 0\) and \(x = 3\) and \(a>0\))