Graphing Quadratic Functions Using Standard Form

This function is written in standard form. To graph a quadratic function written in this form, there are five steps to follow.

1. Identify and graph the axis of symmetry
2. Determine and plot the vertex
3. Determine and plot the $y\text{-}$intercept
4. Reflect the $y\text{-}$intercept in the axis of symmetry
5. Draw the parabola

These steps will be done one at a time.

The Axis of Symmetry

To find the equation of the axis of symmetry, the coefficients $a,$ $b,$ and $c$ should first be found. Consider the given function. Here, $a=\text{-}0.075,$ $b=1.35,$ and $c=0.$ The axis of symmetry is a vertical line with equation $x=\frac{\text{-}b}{2a}.$ The axis of symmetry is the vertical line $x=9.$

The Vertex

The vertex of the function lies on the axis of symmetry. This means that the vertex's $x\text{-}$coordinate is $9.$ Now, to find the $y\text{-}$coordinate, $x=9$ will be substituted into the function rule. The vertex of the function is at $(9,6.075).$

The $y\text{-}$intercept

The $y\text{-}$intercept is given by the constant term $c$ of the function rule. In this case this term is equal to $0,$ which means that the $y\text{-}$intercept occurs at $(0,0)$ — the origin.

Reflecting the $y\text{-}$intercept

Now, another point that lies on the parabola can be found by reflecting the $y\text{-}$intercept in the axis of symmetry.

The third point that lies on the parabola is at $(18,0).$

Drawing the Parabola

Finally, the points will be connected with a smooth curve to graph the parabolic shape. Since the function represents a dog's jump, negative values of the function will not be included.

Since the greatest height that the dog will reach is $6.075$ feet and the height of the fence is $7$ feet, the dog will not be able to jump over the fence. The maximum horizontal distance that the dog will be able to jump corresponds to the distance between the $x\text{-}$intercepts. This distance is $18$ feet. To find how much further the dog will be able to jump, the difference between $18$ and $5$ must be calculated. What a pup!

Vertex Form: $y=a(x-p{)}^2+q$
Sketchtastic	Sugoi Sketch
$y=\text{-}0.015(x-50{)}^2+37$	$y=\text{-}0.008(x-65{)}^2+28$

The given functions can be rewritten in standard form by expanding the squared expression. First, the function corresponding to the Sketchtastic model will be rewritten. By following the same procedure, the second function that represents Sugoi Sketch can also be rewritten in standard form.

	Sketchtastic	Sugoi Sketch
Given Function	$y=\text{-}0.015(x-50{)}^2+37$	$y=\text{-}0.008(x-65{)}^2+28$
Standard Form	$y=\text{-}0.015x^2+1.5x-0.5$	$y=\text{-}0.008x^2+1.04x-5.8$

The functions can now be graphed using the standard form. To do so, there are five steps to follow.

1. Identify and graph the axis of symmetry.
2. Determine and plot the vertex.
3. Determine and plot the $y\text{-}$intercept.
4. Reflect the $y\text{-}$intercept in the axis of symmetry.
5. Draw the parabola.

The axis of symmetry is given by the equation $x=\frac{\text{-}b}{2a},$ where $a$ is the coefficient of the $x^2\text{-}$term and $b$ is the linear coefficient. Use a calculator to evaluate the equations if necessary.

Axis of Symmetry: $x=\frac{\text{-}b}{2a}$
$y=\text{-}0.015x^2+1.5x-0.5$	$y=\text{-}0.008x^2+1.04x-5.8$
$x=\frac{\text{-}1.5}{2(\text{-}0.015)}$	$x=\frac{\text{-}1.04}{2(\text{-}0.008)}$
$x=50$	$x=65$

The axes of symmetry of the parabolas are the vertical lines with equations $x=50$ and $x=65.$

The vertices of the parabolas lie on the corresponding axes of symmetry. This means that $x=50$ and $x=65$ are the $x\text{-}$coordinates of the vertices. Now, these values will be substituted for $x$ into the function rules to find the $y\text{-}$coordinates.

	$y=\text{-}0.015x^2+1.5x-0.5$	$y=\text{-}0.008x^2+1.04x-5.8$
$x\text{-}$coordinate	$50$	$65$
$y\text{-}$coordinate	$y=\text{-}0.015(50{)}^2+1.5(50)-0.5$ $\Updownarrow$ $y=37$	$y=\text{-}0.008(65{)}^2+1.04(65)-5.8$ $\Updownarrow$ $y=28$
Vertex	$(50,37)$	$(65,28)$

The vertices can now be added to the graph.

The $y\text{-}$intercepts are given by the constant term of the function rules.

Function Rule	$y\text{-}$intercept
$y=\text{-}0.015x^2+1.5x-0.5$	$\text{-}0.5$
$y=\text{-}0.008x^2+1.04x-5.8$	$\text{-}5.8$

Next, another point that lies on each parabola can be found by reflecting the $y\text{-}$intercept in the axis of symmetry.

Finally, the corresponding points will be connected with a smooth curve to graph the parabolic shapes.

The $x\text{-}$coordinate of each vertex is the price of the corresponding skateboard at which the profit of the company is maximized. This means that the vertex represents the optimal price and its corresponding profit. The company should use these values to make the highest return on sales.

From the graph, it can be seen that the profits of skateboards Sketchtastic and Sugoi Sketch are about $23$ and $26$ dollars in the thousands, respectively. To verify this, $x=80$ will be substituted in both function rules.

	Sketchtastic	Sugoi Sketch
$x$	$80$	$80$
Substitute	$\text{-}0.015(80{)}^2+1.5(80)-0.5$	$\text{-}0.008(80{)}^2+1.04(80)-5.8$
Evaluate	$23.5$	$26.2$

The estimated values are slightly less than the exact values.

• A player kicks the ball from a distance of $80$ feet.
• If the goalkeeper does not catch the ball, it will hit the goal's line.
• The goalkeeper — standing $15$ feet in front of the goal — catches the ball at a height of $4\frac{7}{8}$ feet.

The given situation can be illustrated on a coordinate plane. Assume that the starting position of the ball is at the origin. The $x\text{-}$axis represents the horizontal distance from the starting point and the $y\text{-}$axis represents the height of the ball. There are two $x\text{-}$intercepts. These are the starting point at $(0,0)$ and the goal's line at $(80,0).$ Therefore, to write the quadratic function representing the path of the ball, the intercept form can be used. The zeros of the parabola are $0$ and $80.$ In the given scenario, the ball does not hit the goal's line. Instead, the goalkeeper, standing $15$ feet in front of the goal, catches the ball at a height of $4\frac{7}{8}$ feet. When the goalkeeper catches the ball, its horizontal distance from the starting point is $65$ feet. This means that the value of the function at $x=65$ is $y=4\frac{7}{8}.$ These values can be substituted into the intercept form to find the missing coefficient $a.$ Finally, the quadratic function that represents the ball's path can be written. To rewrite this function in standard form, the expression $\text{-}0.005x$ will be distributed. The function can now be graphed using the standard form. To do so, there are five steps to follow.

1. Identify and graph the axis of symmetry
2. Determine and plot the vertex
3. Determine and plot the $y\text{-}$intercept
4. Reflect the $y\text{-}$intercept in the axis of symmetry
5. Draw the parabola

The axis of symmetry is the vertical line with equation $x=\frac{\text{-}b}{2a},$ where $a$ is the coefficient of the $x^2\text{-}$term and $b$ is the linear coefficient. The equation of the axis of symmetry is $x=40.$ The parabola's vertex lies on the axis of symmetry. This means that $x=40$ is the $x\text{-}$coordinate of the vertex. Now, this value will be substituted into the function rule to find the $y\text{-}$coordinate. The vertex of the parabola is at $(40,8).$

The $y\text{-}$intercept is given by the constant term of the function rule, which in this case is $0.$ Therefore, the $y\text{-}$intercept is at the origin $(0,0).$

Now, another point that lies on the parabola can be found by reflecting the $y\text{-}$intercept in the axis of symmetry.

The third point lies at $(80,0).$ Finally, the three points will be connected with a smooth curve to graph the parabolic shape. Since the function represents height of the ball, negative values of the function will not be considered.

A parabola that opens downward reaches its maximum at the vertex. In Part B, it was obtained that the vertex of the given function is at $(40,8).$ Therefore, the maximum height of the ball is $8$ feet.

Recall that the $x\text{-}$coordinate of the vertex is $40.$ Since the given distance is greater than $40,$ it can be said that the ball is already falling at $x=50.$

If the goalie understands this math concept, and can apply it in the heat of a soccer match, they are surely going to be an amazing player and maybe some day a sports scientist. What exciting times in what Zain thought was a sleepy town!