Exercise 52 Page 110

To draw the graph of the given quadratic function written in standard form, we must start by identifying the values of a, b, and c. y=-4 x^2-5 ⇕ y= - 4x^2+ 0x+( -5) We can see that a= - 4, b= 0, and c= - 5. Now, we will follow four steps to graph the function.

1. Find the axis of symmetry.
2. Calculate the vertex.
3. Identify the y-intercept and its reflection across the axis of symmetry.
4. Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation x=- b2 a. Since we already know the values of a and b, we can substitute them into the formula.

x=- b/2a

SubstituteII

a= - 4, b= 0

x=- 0/2( - 4)

▼

Evaluate right-hand side

MultPosNeg

a(- b)=- a * b

x=- 0/- 8

RemoveNegFracAndDenom

- a/- b= a/b

x=0/8

0/a=0

x=0

The axis of symmetry of the parabola is the vertical line with equation x=0.

Calculating the Vertex

To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b. Vertex: ( - b/2 a, f( - b/2 a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=0. Therefore, the x-coordinate of the vertex is also 0. To find the y-coordinate, we need to substitute 0 for x in the given equation.

y=-4 x^2-5

Substitute

x= 0

y=-4 ( 0)^2-5

▼

Evaluate right-hand side

CalcPow

Calculate power

y=-4 (0)-5

ZeroPropMult

Zero Property of Multiplication

y=0-5

SubTerm

Subtract term

y=-5

We found the y-coordinate, and now we know that the vertex is (0,-5).

Identifying the y-intercept

The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Therefore, the point where our graph intersects the y-axis is (0, - 5). As we see, the y-intercept is the same as the vertex, then it will not help us to draw the graph. Instead, we will find the y-coordinate of the points where x=- 2 and x=2. Let's start with x=- 2.

y=-4x^2-5

Substitute

x= - 2

y=-4( - 2)^2-5

▼

Evaluate right-hand side

NegBaseToPosPow

(- a)^2 = a^2

y=-4(2)^2-5

CalcPow

Calculate power

y=-4(4)-5

MultNegPos

(- a)b = - ab

y=- 16-5

SubTerm

Subtract term

y=- 21

Now, let's find the value of y when x=2.

y=-4x^2-5

Substitute

x= 2

y=-4( 2)^2-5

▼

Evaluate right-hand side

CalcPow

Calculate power

y=-4(4)-5

MultNegPos

(- a)b = - ab

y=- 16-5

SubTerm

Subtract term

y=- 21

Let's now plot the points we already have.

Drawing the graph of the function

We can now draw the graph of the function. Since a= - 4, which is negative, the parabola will open downwards.

We can see above that the maximum point of the curve is reached at the vertex.