Question

Find the equation of the parabola through the point \((6,-5)\) if its vertex is at the origin and its axis is along the \(y\) -axis. Make a sketch.

Short answer

The equation of the parabola is \( y = \frac{-5}{36}x^2 \).

Step-by-step solution

Determine Parabola Equation Form

Because the parabola has its vertex at the origin (0,0) and its axis is along the y-axis, the equation of the parabola can be expressed in the standard form: \( y = ax^2 \). We aim to find the value of \(a\).

Substitute Point into Equation

We are given the point (6, -5) that lies on the parabola. Substitute \( x = 6 \) and \( y = -5 \) into the equation \( y = ax^2 \): \[ -5 = a(6)^2 \] This simplifies to: \[ -5 = 36a \].

Solve for Coefficient 'a'

To find the value of \( a \), divide both sides of the equation \( -5 = 36a \) by 36:\[ a = \frac{-5}{36} \].

Write the Equation of the Parabola

Now that we have the value for \( a \), substitute it back into the equation \( y = ax^2 \). Therefore, the equation of the parabola is:\[ y = \frac{-5}{36}x^2 \].

Sketch the Parabola

To sketch the parabola, note the vertex is at the origin (0,0), and because \( a \) is negative, the parabola opens downwards. The point (6, -5) lies on this parabola. Plot the vertex and point, then draw a symmetric curve around the y-axis passing through these points.

Key concepts

Parabola Equation

The equation of a parabola is a way to describe its shape and position on a graph. Parabolas are U-shaped curves that can either open upwards or downwards (vertical orientation), or sideways (horizontal orientation). In mathematics, the equation of a parabola can be written in different forms, with each form providing specific information about the parabola's characteristics.

• Standard Form: The most basic form of a parabola equation is the standard form, which is written as \( y = ax^2 + bx + c \) for vertical parabolas. For horizontal parabolas, it would be \( x = ay^2 + by + c \).
• Vertex Form: This form highlights the vertex of the parabola, expressed as \( y = a(x-h)^2 + k \) for vertical parabolas, where \((h, k)\) is the vertex.
• Factored Form: Useful for finding roots, this form is \( y = a(x - r_1)(x - r_2) \), describing where the parabola intersects the x-axis.

In our exercise, because the parabola’s vertex is at the origin and the axis is along the y-axis, the simplified form is \( y = ax^2 \). This is because the vertex form \( y = a(x-h)^2 + k \) reduces to \( y = ax^2 \) when \( h = 0 \) and \( k = 0 \), thus focusing only on the coefficient \(a\). The value of \(a\) determines the width and direction of the parabola.

Vertex at Origin

The vertex of a parabola is a point indicating where the curve reaches its highest or lowest point. For vertical parabolas, this could mean the bottom of a "U" or the top of an upside-down "U." When the vertex of the parabola is at the origin, its coordinates are \((0,0)\). This is a special scenario because it simplifies the equation.

• Standard Case: Normally, the vertex form of a parabola is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex.
• Simplification at Origin: When \((h, k) = (0,0)\), the equation simplifies to \( y = ax^2 \) for vertical parabolas.

Since the vertex is where the direction of the curve changes, when the vertex is at the origin, the parabola’s equation reflects that symmetry very clearly. This contributes to understanding the overall shape and position of the parabola without adding other constant terms that shift the graph horizontally or vertically.

Axis of Symmetry

The axis of symmetry in a parabolic equation is an imaginary line that divides the parabola into two mirror-image halves. For vertical parabolas, this line is vertical, and for horizontal parabolas, this line is horizontal. For our specific exercise, where the parabola’s vertex is at the origin and its axis is along the y-axis, the axis of symmetry is essential in determining the parabola's shape direction.

• Vertical Parabolas: The axis of symmetry is a vertical line \(x = h\), but if the vertex is at the origin \((0,0)\), then it’s simply \(x = 0\).
• Orientation and "a": The value of \(a\) determines if the parabola opens upwards or downwards: a positive \(a\) means upwards, and a negative \(a\) means downwards.

The axis of symmetry helps in graphing by providing a reference to ensure both sides of the parabola are equidistant from this line. Understanding this concept allows one to sketch a perfect, symmetrical U-shaped curve, knowing that each point on the parabola has a twin point on the other side of the axis.